 All right, so good afternoon everyone. Welcome to the first session on metasis now People who have already done this in school would be should be able to answer this question. What do you understand from a matrix? How do you define a matrix? So we all know it's an array of numbers, right? It's an array of numbers and In fact, it's an array of some physical quantity, right? Why I'm saying physical quantity is because it could be Scalar quantity, it could be vector quantity. Correct. It could also have Complex numbers, real numbers as its elements. So what is a matrix? Matrix is nothing, but It's an array It's an array It's an array of physical quantities Physical quantities Okay, right and these quantities which are also called as the elements of the array or members of the array Right, they could be vectors. They could be scalars Okay, so these these physical quantities can be vectors Okay, they could be scale as well. Okay, so let's not have any kind of a doubt that only Metises can be formed out of scalar quantities. It can be found out of vector quantities also. Okay, and Not only that your You know quantities could be real numbers or it could be imaginary numbers or non real numbers also Okay Now, why do we need a matrix? Why do we need a matrix? Why do we need a matrix? Let's say if I give you a matrix like this Let's say two zero Zero three Okay, what do you understand from this? anybody Why does actually used is it just used to maintain a list Having some rows and columns right, for example, if you know My mom gives me a certain grocery item to purchase So she will give me the name of the item how many kgs I need to buy and probably she will you know Give me the money as per what is the rate of those substances? So is that what an array is what an matrix is meant for as a Science student as an engineering Students who do you do you think that is the application of matrices for you? Why do we need matrices? What do you understand when somebody tells you that there's a matrix like this? What what do you understand from it? Yeah, take it take a guess take a guess I It's an organized data. Okay, so do you think a science student needs to know how to organize data without any application to it? Okay, now, this is a drawback that we most of us carry when we don't understand why we are studying matrices See matrices are basically agents of linear transformation. So let me write it down very clearly over here matrices are matrices are agents of Agents of linear transformation agents of linear transformation Okay, what are the meaning of the word linear transformation? This is a very difficult word to digest in first go see when I see this basically I see my 2d space right Transformed in such a way That see what is a 2d space 2d space is made up of x and y Right What are the basis vector for x-axis? What is the unit vector for x-axis so as to call it? I right I cap most of you would have done it in physics I cap right and What are the basis vector for this? J cap right see what is basis vector basis vector is basically the vectors which you replicate to generate the entire 2d space Correct. For example, if I want to generate a point Let's say 2 comma 3 Okay, basically it is generated when I move two units along the x-axis and probably three units along the y-axis Right, are you getting my point? So any vector in this is basically a position vector. You can call this as a position vector Okay, so any vector in your 2d space or for that matter if I give you a 3 by 3 it would be a 3d space that is obtained by Basically some basis vectors or unit vectors what we call it Okay Now when I say I'm basically making a matrix like this now This is basically a matrix which changes your unit vector along the x-axis by two times and Unit vector along the y-axis by three times that means instead of having a one by one grid It transforms this space in such a way that now your grid becomes two by three grid Are you getting my point? So it is linearly transforming the space Okay, now linearly transforming the space means it is trying to Change your basis vectors, which is making the entire space and When it is doing that it basically Changes every vector lying on this space. So if let's say if I had a vector 2 comma 3 Then this vector would have changed it to I think most of you would have done matrix multiplication It would have changed it to 4 comma 9 So this point 2 comma 3 in this new system will now go to 4 comma 9 Are you getting my point? Okay, I'll show you something very interesting over here Sir is it like shifting of origin? No, no, no origin doesn't shift only the Basis vectors they stretch or they contract Right, they may they may become oblique also, but there is no shift of origin Right. So linear transformation just changes your x comma y to ax plus by kind of a vector I'll just show you something very interesting. You'll be able to relate to that. Let me open my Geo Gebra Okay, so All of you please pay attention over here. I'll show you something very interesting Let's say this is my 2d space Correct. Okay. Now what I'm going to do is on this 2d space on this 2d space I'm going to drop a picture. I'm going to drop a picture. Okay, let's say Yeah By the way, she's my daughter. So I'm taking a picture. Okay, and I'm dropping it over here Okay Fine Now every pixel in this image Every pixel in this image is basically a position vector. For example, her eyes will have a position vector Right. Her nose will have a position vector. Okay, so every No, you can say every spot on this image is having a position vector Okay. Now, let's say I want to transform this linear space. Let's say I want to make her fatter Okay, so see what I'll do Let me just rename her pick something. Let's call it as P okay So what I'm going to do is I'm going to now make a factor through the use of Matrices so I will write a vector or I'll write a matrix F F for fat. Okay See what I'll do. Okay. As you can see I've written some matrix Fine. Now we'll apply this matrix F to her image P See what happened You see that her image became distorted Now what happened? Basically, I wrote such a matrix. I wrote such a matrix over here which Which linearly transformed this space in such a way that any position vector on it Let's say any position vector x comma y on it After getting linearly transformed by this matrix became 2x y that means It became elongated twice along the x-axis, but it did not allow elongate along the y-axis Are you getting my point? Yes, sir matrices are Are a very very important tool in field of digital image processing artificial intelligence A lot of field matrices are going to be are being used now. I come from electrical background So I use it in image processing. Okay So whatever tools you use know your picasso photoshop Or let's say after taking a photograph you want to increase the contrast or you know Try to make yourself look, you know a little bit taller or you know They're all done through matrices actually working at the backdrop Now, let's say I want to flip her image about the about the Let's say y-axis. What will I do? Tell tell me anybody can minus one But x replace x and y with minus one excellent very good. See now what I'll do is I'll make a I'll erase this I let's say take another Let's say g And as Anjali rightly said I'll make a vector Now I have to change I have to reflect her image about the y-axis means I have to change x with minus x Right, so I'll make a vector like I'll make a matrix like this see Okay Now I'll apply this I'll apply this to I'll apply g to her image b see It got mirror image I can do anything I can make her taller by stretching it right Can you tell me something to reflect her image about x-axis what will I do tell me Same thing but Okay, one zero and zero minus one correct Zero minus one and apply this minus one sir. Oh, I'm so sorry Apply this to Apply this matrix t to her image. Okay, as you can see it has now gone down Okay Can you see that? Okay, you can also know rotate her image Right, you can also, uh, you know take the mirror image of the you know, whatever transformations you can think of you can apply it To the use of matrices. Okay, so we'll talk about these applications in the last part of matrices I think in the last class of matrices, which is which which might be the next class But remember my dear students matrices are not there for you to arrange data It is not there for you to make a list of what grocery items you're going to purchase They have scientific applications And they are very powerful tools very very powerful tools Okay So let us begin with this understanding about matrices. So from now onwards, you should not take it very lightly that okay You can go home and do it on your own. It has got very tricky concepts, which we'll discuss in today's class also First we'll talk about types of matrices early. This part will be very easy for you But it is important because uh, when I'll be referring to these matrices, you should know their names Okay, so what are the types of matrices? We'll first start with the row matrix I'll just be quick over here because we all know this. So what's the row matrix? Which has got only one row in it and it can have any number of columns. So it has one row Right, but it can have any number of columns And any number of columns Uh to take an example something like One five seven. This is a row matrix Okay column matrix. What's the column matrix? Having only one column Only one column and any number of rows Any number of Rows Sir, what would happen to our image if there were no zeros at all? Uh, what is the meaning of zeros there? Uh, site Every image every element of her pixel is basically a position vector. So when you're changing your space, let's say you are expanding it along x axis or y axis Any kind of squishing or you know, whatever you do with this space It will be applied to every vector on it. So what is happening? Her image is getting distorted because of that Oh, okay, sir. Are you getting my point? Yes, sir. Yeah So matrices are basically changing the space so that everything on that space is getting distorted So image processing is one of the you know, um Very very commonly used application of matrices. Of course data data analysis also involved But that's because I come from you know circuit background. I was giving you idea about uh image probably if you talk to Uh person who has an artificial intelligence like Tushar sir He will give you a different idea altogether about you know the use of matrices But I what I wanted to say it's not it's not a simple, you know Just listing down of numbers in in rows and columns. It has much more uses to it Yeah, so column column matrix example would be something like one five seven Normally we uh write our vectors having three dimension as a column matrix Okay, next is Next is basically a rectangular matrix Just know the names. It is not that it's going to be very very important for school or this thing So what's the rectangular matrix? Now, how do you represent a matrix? I think I should have told you that first so matrix is basically represented by A capital alphabet a or b or whatever right and we write It like this a i j basically signifies the element of that matrix present in i th row and j th column Okay, m signifies the number of rows in the matrix and signifies the number of columns in that matrix You are already aware of these you know concepts in your junior classes. Okay, so A rectangular matrices are those matrices where The number of rows are not equal to the number of columns Now there are two types of rectangular matrices One is basically horizontal matrix. Let me write it in different color horizontal matrix Okay in horizontal matrix Basically the number of Columns will be more than the number of rows Okay, so here the number of columns would be more than the number of rows For example, something like this is a horizontal matrix one two three four five six Okay, as you can see this is a two by three matrix number of columns are more than the number of rows Okay, there can be something called vertical matrix Vertical matrix Okay, so vertical matrix basically is one where your number of where your number of Rows will be more than the number of columns So something like this will be a vertical matrix one two three four five six So number of rows would be more than the number of columns Okay Now The important one for us is this matrix called square matrix Square matrices would be the most commonly used matrices by us. I think you have done determinants also nps aja ji nagar Determinants are found only for square matrices Okay, in fact the special type of matrices which will which will study today like symmetric matrix q symmetric Uh hermitians q hermitian. They're all square matrices only Okay, so what's the square matrix? Basically where your number of rows and number of columns are equal That's a square matrix Okay in square matrix One thing that we all need to know there's something called the Leading diagonal or the principal diagonal. Let me take an example of a square matrix. Let's say three two one two three one Okay This diagonal is basically what we call as the leading diagonal Leading diagonal Okay, also called as the principal diagonal principal P a l okay principal diagonal Okay Now for a square matrix For a square matrix the sum of the elements in the leading diagonal is called the trace So what is that trace? Trace of a square matrix is nothing but it is the sum of the elements Sum of the elements in the leading diagonal We'll talk about trace in some time Okay, some of the elements in the leading diagonal is called the trace fine Any any question about the square matrix? Why should determinants be only found for square matrix? We'll talk about it. We'll talk about it Determinants. Yeah determinants also have their own physical significance just like matrices have So determinants give a number that basically tells you that How is your see when you have a one by one square in a in a grid? Okay When you make a transformation on that linear space Then the same unit you can say unit parallelogram How would the area of that getting changed because of that transformation? That factor is basically told by the You know determinant value I would request all of you to watch this video by three blue one brown Sorry, three brown one blue or three blue one brown something like that Three blue one brown Three blue one brown yeah They have given a very very you know pictorial and animated view of how vectors and determinants are to be perceived In 11th and 12th we get a very very wrong picture about matrices and determinants And let me tell you this chapter is again going to come in your first year of engineering Okay, so Matrices are going to come again in first year of engineering All right next is Null matrix Null matrix or zero matrix Okay, what is a null matrix or a zero matrix? basically A matrix of order Any order let's say m cross n Says that all elements of this matrix are zero Okay Normally we represent this matrix by o Okay, don't confuse it with zero it is o Okay represented by o So this plays the same. I know a role that uh Basically a zero number will play in terms of our common, you know use numbers Okay, that depends upon the space which you are dealing with next is diagonal matrix What's a diagonal matrix now nps? I would like uh you to tell me what have you learned about a diagonal matrix What has been told to you about the diagonal matrix? Sir, so only in the principle diagonal there'll be elements and everything else should be zero Everything else should be zero. Okay. Now many books will write this definition By the way diagonal uh matrix are basically called sparse matrix Because they have zero in most of the places and as sahana rightly mentioned The elements are only present in the diagonal position or you can say non zero elements are only present in the diagonal positions Rest all the elements are going to be zero. So basically A diagonal matrix is first of all a square matrix First of all a square matrix Okay, and it follows It follows this algorithm that if i is not equal to j the element would be zero correct okay, and If i is equal to j the element should not be zero Now this is slightly controversial. Many people uh, they claim that The elements in the diagonal position May be zero Okay, but element in the non diagonal position must definitely be zero So i've seen this controversy in many places in kora also Mathics stack exchange also people are debating about it But see the idea here is that Normally in a diagonal in a diagonal matrix People say that these three elements Should not be zero And rest of the element should be zero, but this is a this is a controversial thing However, it has not hurt anybody because of this controversy. Okay But if you want it to be invertible then none of the element should be zero Now what is invertible matrix purely nps raja ji nagar would be aware of it, but not Yeshwantpur, are you aware of what is invertible matrix? Yeah, she she taught invertible matrix Okay Now i'll tell you some very good characteristic about a diagonal matrix diagonal matrix Because there are a lot of sparse, you know values over here. That means the values are zero. It is represented as D i ag and when you just write your elements, let's say a b c It is obvious. It is very, you know clear that it will only have a b c in the diagonal positions Okay, so many a times, you know, you don't have to write all unnecessary zeros Now the characteristic of a diagonal matrix, let me name it as let's say a diagonal matrix d Okay, let's say i'll take an example of a diagonal matrix Whose diagonal positions are a 1 a 2 Till a n let's say it's a n by n square matrix, which is a diagonal matrix The very important characteristic is If you raise it to any positive integer powers the direct impact can be seen on the elements Okay Let me not choose n. Let me choose k so that We don't mix it with the order Okay And this is applied even if You are inverting the matrix. So if you want to reciprocate or if you want to the reciprocate word is basically synonymous with If you want to invert the matrix, okay, of course, we don't do one by the matrix There is a separate operation which we call as the inverse finding the inverse of the matrix then if you are finding the inverse of the matrix the diagonal elements would simply get inverted so it'll become one by a 1 One by a 2 and so on till one by a n Okay, this may not be shown by all the matrices. This may not be shown by all the matrices Okay, so Aniruddha has asked what is the significance of diagonal matrix. We'll talk about you know diagonal Sorry diagonal leading diagonal So there is something called eigenvectors and all which we'll study in some time there. You'll understand the importance of the leading diagonal Aditya, which which which part do you want me to go to? No, sir In case I in case I forget your forget to see the chat. Please do remind me by speaking out. Okay Okay, so Venkat has a question sir. Can the matrix have non zero elements in the non leading diagonal and zero in the No, no, no, no See Venkat, this is the definition which which basically is given By many of the authors which I have seen for example, this is the definition given by Amit Agrawal Okay, Amit Agrawal says the leading diagonal elements should not be zero But some books I'll just write out a way some books And in some, you know, you can say um, you know websites um Let me write sources They say that the non leading diagonal positions must be zero Okay, that is what they write. They don't say anything about the diagonal or leading diagonal elements Okay, they may or may not be zero is what they say Okay, so that there's just that's why I asked NPS Rajaji Nagar. What has been told to you in school? So we wasn't told anything we learned it ourselves Okay, what does NCRT state? Yeah, I think the first definition This one Yeah Okay, because many books I have seen even uh, I have read through the books of Cengage and all they just say the non leading diagonal should be zero What about leading diagonal element? They may or may not be zero. Nothing has been specified about them Okay So this is something which basically is a peculiar characteristic of a diagonal matrix In fact Later on you learn matrix polynomials Right. What is matrix polynomial? I'll talk about it. But let's say if you are finding a I'll just give you an example This is a very interesting property of diagonal matrix see F here is a function. Let's say it's a polynomial. Let's say I give a polynomial like this, um x square plus x plus two Okay Right, so if you If you replace your x with a diagonal matrix Then what happens? Every element of the diagonal matrix in the leading diagonal position will undergo the same treatment Okay, for example, if your diagonal matrix was I'll just give you an example of You know something like this One two Okay, so if you want to find f d Okay, then basically what will happen is your you'll again get a diagonal matrix whose elements will be one square plus one plus one Which is four and two square plus two plus two which is how much eight, right? Are you getting my point? Okay, so as to say that if you're doing d square plus d plus two i Then you'll end up getting this expression Okay We'll talk about matrix polynomial in some time when we are talking about multiplication of matrix But since I was talking about diagonal matrix, I thought I would introduce you to these properties Is it fine any questions here anybody? okay now A special type of diagonal matrix is basically a scalar matrix Let me write it down scalar matrix What is a scalar matrix? Scalar matrix is basically again a square matrix Number one. It should be a square matrix Okay And the property of this matrix is let's say I talk about a square matrix of order n By the way, when I say square matrix of order n means What is the number of rows and what is the number of columns? And then each right, okay So it's just like saying that a is a I mean just I'll let it I'll write it down over there. It just implies that a is basically a matrix of this nature Okay So scalar matrix basically satisfies this property the element satisfies this property It would be zero if i is not equal to j And it would be k if i is equal to j and k should not be zero And k should not be zero Okay, so a typical example of this could be three three three zero zero zero zero zero zero Okay, so as you can see it's a diagonal matrix But having the same element in the leading diagonal position. So these are all elements are same Okay, this is a matrix which you will realize when we are doing A joint of a matrix into a Okay, we'll talk about it when we are doing determinants. In fact, when we are doing, you know, adjoined of a matrix also Okay A special type of scalar matrix where k is one is basically called a unit matrix Or identity matrix So next is a unit matrix or Identity matrix Okay, so what is this basically again? It's a square matrix It's a square matrix And it is a special type of scalar matrix where All elements in the non-diagonal positions are zero And all elements in the leading diagonal positions are one Okay, it plays the same role as the identity element. Now, what is an identity element? Uh, n ps y pr. Have you done binary operations? No Okay, see identity elements are basically those elements of that given set Which when you know applied Now they can be a binary operation. For example, let's say zero is an identity element for addition On real numbers. So if you take any real number add a zero to it. It'll give you the same real number, right? So we say zero is we say that zero is an identity element For addition Identity element for addition on real numbers For the binary operation of addition on real numbers on real numbers Similarly, uh, one is an identity element for multiplication on real numbers, right in the same way I is basically an identity element for multiplication Okay on square matrices Okay, so if you if you are applying Let's say multiplication on two square matrices Then on that type of operation. I will play the role of the identity matrix or identity element Okay, just like oh will play the role of identity element for addition Between two matrices of the same order We'll talk about it when I talk about addition also Okay Normally we represent it by i n Where n specifies the order Okay, so for example i2 if you say it means 1 0 0 1 this is i2 I3 if I say it means 1 0 0 0 1 0 0 0 1 Okay So this is the identity matrix of order 2 identity matrix of order 3 depending upon you can go on to any n by n matrix Next we'll talk about Operations on matrices Operations on matrices by the way the list is not over Uh, there are some special type of matrices which we'll understand in some time after we have done some operations Because they are known from the operations done on them. I don't know whether you have been introduced to a nil potent involutary Item potent all those stuff. Have you been introduced nps? I think potent is no I don't think so Okay, okay. No no problem. We'll talk about it. We'll talk about it So let's look at some operations on uh uh matrices The first operation is equality of two matrices equality of two matrices so when you say When you say matrix a is equal to matrix b When you say matrix a is equal to matrix b Okay, it can only happen It can only happen if The number of rows of a is equal to the number of rows of b number one number of rows Of a should be equal to number of rows of b Okay, and secondly number of columns Of a should be equal to number of columns of b Okay, in other words, they should be of the same order Okay, that is a and b must be of the same order Must be of the same order. This is one criteria Okay, so let me call this as criteria number one second criteria is every element of a Must be equal to the corresponding element of b Right So when each of the elements of a and b are correspondingly equal to each other Then only we can say the two matrices are equal Now there is no relation like met one matrix being greater than or less than other matrix Inequality properties do not hold good. Okay So just a simple example to take if somebody says if somebody says uh, let's say x y z w Is equal to x six minus one to w And I ask you what is the value of these elements x y z w and all what will you say? You can't find out Or just once again, I'll I'll give you a value for this. Let's say four W is zero Yeah, so in this case you will compare element with elements You'll say x is equal to four y is equal to six z is equal to one and since w is to w means w is equal to zero Okay, so only by comparison you can basically find out the values of these elements. Now, this is a very simple concept Let's not waste too much time on it Next we'll talk about Next we'll talk about addition of addition of matrices addition of matrices Now, let's say when you're adding two matrices a and b They can only be they can only be added to each other or what we say in the language of mathematics as that they are confirmable to be added Or confirmable for addition Okay, so if a and b are confirmable for addition it can only happen When number one order of a should be equal to order of b Okay, and secondly, let's say if you get an answer c Then c will also be of the same order as a and b Okay, so in addition the the outcome is also of the same species as whatever has been added For example mango plus mango can give mango only it cannot give us banana, right? and secondly Every element of c Every element of c is made by the sum of the corresponding position elements of matrix a and b for example Let's say first or second column element is obtained by adding first or second column of a With the element present in the first or second column of b Okay, a simple example I would like to give air. Let's say we have one two three four And let's say you are adding it to minus one six seven nine Okay, so first of all are they of the same order? Yes, then only we can add it and how they're added you basically add element to element So this is zero eight 10 13 Okay, nothing very great about it Some properties we will have to know about addition of matrix Let's look into the properties Matrix addition If you feel I'm going slightly faster, please stop me. Okay The concept is very easy. That's why I'm moving in a slightly faster pace. Why not? We can have a one by one diagonal matrix. Why not? diagonal matrix is basically a square matrix It can be one by one two by two three by three whatever Got it done Okay, so what are the properties of matrix addition number one? Matrix addition is commutative That means it doesn't make a difference whether you are doing a plus b Or b plus a the result is the same. Okay provided. They are confirmable for addition fine number two Matrix multiplication is associative What is the meaning of associative? If you do a Yeah, if you do a plus b and then add it to see it is same as doing a plus b plus c It doesn't make a difference to it Okay, then there is something called the existence of existence of additive identity Identity The null matrix of the same order as a or b. Okay So what is an what is a additive identity for a matrix addition? It's basically a matrix o Which is of the same order Same order as you know a for example, let's say a plus o Is a then this o will be a null matrix of the same order as a let's say a is one two three four five six Then the additive identity for this would be zero zero zero zero zero zero Are you getting my point? Okay, so I don't think so there would be any question on this but if let's say they ask you What do you think is the additive identity for this matrix? Okay, don't write any order null matrix It should be of the same order next is existence of additive inverse existence of additive inverse So what is the additive inverse of a matrix a so basically a matrix? Which added to that gives you the additive identity for the same matrix? For example, if I say Uh for this matrix a what do you think is the additive inverse? minus a All right Yeah, so this would be the additive inverse of a So as you can see additive inverse of a would be Minus of a so in this case in our given example It would be minus one minus two minus three minus four minus five minus six Okay, minus of a means you just change the sign of every element of that given Given matrix basically it comes under the concept of scalar multiplication, which I'm going to take next after this Next property is The cancellation laws hold good the cancellation laws hold good What are the meaning of cancellation law hold good? See if I say a plus b is equal to a plus c okay right Normally from here we can say b is equal to c This is called left hand cancellation law or left cancellation law Left cancellation law Okay, because you can cancel the left hand terms the terms to the left of these terms. So whatever our left will be equal Okay Similarly, if I say b plus a is equal to c plus a then it can be concluded that b is equal to c By cancelling a from the right sides of these terms. So this is called the This is called the right cancellation law So both these right cancellation and left cancellation law, they hold good for matrix addition But you will learn in some time that they do not hold good for multiplication Okay, now things may seem very trivial. Oh, sir, this is very obvious. No, why we are learning this It's not very obvious matrices. They don't follow the same, you know Uh laws and rules as normal numbers follow. Okay, you'll see in some time. What are the exceptions that Matrices show Okay Now i'm not covering a subtraction because subtraction is basically adding one matrix to the negative of the other So i'm not covering it separately. I'll directly jump now to scalar multiplication Any question with respect to this? So next is uh scalar multiplication Scalar multiplication basically when you have a matrix a let's say Of order m cross n Right And if you take k as a scalar quantity And do this k a Okay, this is called scalar multiplication. What happens in scalar multiplication All the elements of the matrix get multiplied with that scalar quantity k Okay, for example, let's say if your a is one two three four five six And if I say Give me two a then what are you going to do? You're going to multiply each and every element of this Uh matrix by two so it'll become two four six eight ten twelve Correct now this property is slightly different from what we follow in determinants I don't know whether you have done determinants, but in determinants only one row or one column could be multiplied with k Okay In a in a determinant when you multiply by a scalar quantity k only one row or only one column Of your choice Can be multiplied with that k But in matrices all the elements of the matrix have to be multiplied with that k Is that fine? Okay, some uh small properties we can uh List it out As notes properties of scalar multiplication Initial part is very easy uh Venkat Aja many people take this chapter very very lightly. Let me tell you two questions will come in j main Two questions will come in j main and two questions will come in j advance on this topic Okay, so not to be taken lightly eight marks Will be you know definitely coming from this topic All right, so if you multiply uh Some of two matrix a and b with a scalar quantity lambda then it is as good as multiplying them separately with lambda and adding it Okay, and let's say if you have two scalar quantities, then you basically can write it separately also like this These are all simple properties to deal with so if you have lambda multiplied to let's say mu Mu a you can write this as mu Mu times lambda a that means you can apply some kind of you know associative property over here Okay, sir. Yes. What happens when you add a scalar to a matrix? You can't add it. No Okay, thank you Addition will always happen between two quantities of the same nature It's like asking what will happen when you add bananas to apples You can't add them Okay Yeah, so quantities of the same nature can only be added So one doubt regarding cancellation law Yes, sir, isn't it just redundant because it's commutative either ways Redundant means like uh, whether the law exists or not it won't matter, right? The cancellation law See, uh, when you when you cancel out some stuff, basically we try to take take the root what we follow in numbers Okay, and in case of addition. Yes as you are right. It is commutative It is going it's not going to make much of a difference But where commutativity fails for example in matrix multiplication there it may matter to you Right So yes, okay, sir And if you're right that because of the because of the fact that they are commutative The cancellation law basically hold true Okay, but when you study linear algebra to a higher level I think you'll be studying in undergraduate. You would realize that rules are very tricky there Okay, what we normally apply for uh Numbers may not work for matrices fine Yeah, so last property as you can see if you do minus Scalar quantity into a doesn't matter whether you treat it any any one of these ways All of them are treated in the same way Next we'll go into some serious part of this chapter. These are all very simple matrix multiplication of matrices multiplication of Matrices This is very important Because the major property of transformation is basically hidden in this particular, you know operation So when you're transforming a matrix Uh, when you're transforming a scalar Vector quantity or a position vector or whatever You are basically doing a pre facto or a post facto multiplication with a matrix Now, how is the matrix multiplication done? Most of you are already aware of it So let's say a is a matrix which is of order m cross n And b is a matrix of order I'm just taking some, you know um Random order let's say p cross q Okay If you are multiplying these two matrices Then the conformability test for this is that the number of columns of the Pre facto so when you're multiplying a b a is called the pre facto b is called the post facto Okay, so the number of row Columns of the pre facto must be equal to the number of rows of the post facto Right, so the conformability test is So what is it called? This a Yeah Pre facto pre facto post facto Pre facto post facto Okay And so the conformability The conformability for Multiplication is Multiplication is The number of columns of the second Should be equal to the number of rows of the first Okay, that means n and p should be the same. So let me re rephrase it now. So a is basically let's say a i j m cross n then b should be b i j n cross q Okay, and the resultant matrix c c would be of the order m cross q that is m here and q here that would become the order of the resultant matrix Now, how do you find the elements of the resultant matrix? I'm sure your book has Mentioned a summation process in this case, which is nothing but a i k into b k j where k is basically Starting from sorry starting from one and going all the way till the number of Rows of the second or columns of the first. So basically it goes from one to n Okay, and i basically goes from one to n i goes from one to n and j will go from One to q Okay Now let me show you an example how it works Let's say This is your a and this is your b Are they confirmable to be multiplied? Yes, it is a two by two and this is also two by two Okay, so how does matrix multiplication happen all of you please pay attention those who have not done this before What do we do we start with? Follow the symbol So take the row of the first multiplied with the column of the second. So two will multiply to five One will multiply to one and they will get added up Okay, so if you take the first row of a and first column of b Then whatever answer you are writing you're going to write in the first row first column position So if you take the first row and second column You will write the answer in the first row second column of c like this Similarly, if you take the second row and first column then you will write your answer in the second row first column of c Okay, so the idea is clear whichever row you are taking from a And whichever column you are taking from b After multiplying the corresponding elements, you have to write it in the same row column position of the answer matrix So this will come three into two plus five into seven Okay If i'm not mistaken, this is 11. This is also 11. This is 20 and this is going to be 41 So this will become your You know resultant matrix Are you getting my point here? Okay, now some properties Which are very very important and direct questions can come on them in Our comparative exams Let's look into that. Let me let me write it in the previous page on this So actually I had a doubt. Yeah, so if we have to find out like the cube of a matrix So first we find out a square and then do we To find out a cube do we multiply a square with a or a with a square? Very good. Your your answer would be Obtained in this property, which I'm going to tell you Okay So properties of matrix multiplication properties of matrix multiplication Okay, the first property is In general matrix multiplication is not commutative So It's not commutative Let me write it like this not commutative in general in general means In some cases they may be commutative, but generally they're not commutative. That means ab Need not give you same answer as ba Okay And especially in those cases where you know your orders are you know in such a way that if you switch the positions The conformability condition may get violated Okay, so in those cases, of course Ab and ba will not give you the same result. Okay Now even when a and b are square matrix This result may not hold true Now coming to Anjali's question When you do a to the power n First of all, let me tell you what is the meaning of this? It means of course n has to be a positive integer Okay, it means that you're multiplying a to itself n number of times a to the power n doesn't mean you're raising every element to the power n No, that is not the meaning of a to the power n a to the power let's say if I say a to the power 3 it means I'm multiplying a with a with a Okay Now the next property is going to help us I know answer question of Anjali. Our question was if I do a cube Whether do you do a square into a or a into a square? Okay So the answer is hidden in the fact that matrix multiplication is associative What is the meaning of associative? Associative means if you do a b and then multiply with a c it is not going to Change even if you do this provided, of course, they should be all confirmable to be multiplied Okay Now to answer Anjali's question a cube is basically a into a into a so whether you do this Or whether you do this Your answer is not going to change So whether you do a square into a or whether you do a into a square In this case your answer is not going to change because of this associative property Did you get your answer Anjali? No. Yes. Thank you, sir It doesn't matter whether you do a square into a or you do a into a square and this would be the same It is because they follow associative property Now in associative property one thing I would like to add over here So let's say if a is m cross n b is a n cross p And let's say c is p cross q Right, then the resultant would be let's say a matrix which will be of the order m cross q Okay, so the rows of this and the columns of this would be the resultant matrix So the same trend has been followed Okay Yes, Venkat. You're saying something Yes, I had it out. Yeah. Yeah Could you go up send the commutative property please? Yeah, sure Yes, sir. So we know that multiplication identity is one right unit matrix of the same order Yes So uh, you know like Unit matrix is always square matrix Yes, sir. Yes So in order to make the um matrix multiplication associative can't we just multiply both sides with uh unit matrix Not unit, I've got just a matrix full of 1s No, no, no, you're not getting the fact here unit matrix term itself is applied only to a square matrix unit matrix is always a square matrix Oh, okay So if you're multiplying anything to a unit matrix make sure it is confirmable first of all to be multiplied Okay, it is not like it is not like you know a number that you can multiply with anything to get the same answer No, it must be confirmable first of all to be multiplied Yeah, now what is your question? You got your answer? Yes. Yes. Okay Sir, even I had a doubt Yeah, yeah, tell me Sir, you said uh, what if you exchange it like that then m and q will change yourself No, it'll not change. No, that's the point if you multiply a and b first As you can see the number of columns here is same as the number of rows over here, right? And if you do this, let's say bc the number of columns here will be same as the number of rows here Correct. So confirmability test is not disturbed Got it Yeah, sir Yes, sir. If all of them were transpose matrixes matrices, then also would this uh rule be applicable What are the meaning of transpose matrix? There's a transpose of a matrix What is transpose matrix in itself? Sir, is it like a rose with the more columns? So you find transpose of a matrix, right? Yes, sir There's no matrix called transpose matrix. There's nothing like inverse matrix. There's the inverse of a matrix is there Correct. No So what is your question if you transpose a matrix then you get another matrix then what do you do with that? So the same thing like uh, how you wrote here now, sir a a uh, I mean like Trans if you find transpose of a b and c would Still give d transpose d transpose of d m into q Okay, now the order will be different in that case. It'll go from the reverse direction We'll talk about it under those properties. Don't worry. Today. We are going to talk a lot about transpose properties Okay, okay, sir. So what you are what you're talking is basically something which we call as law of reversal Okay, we'll talk about it. We'll talk about it. Let's go step by step. Let's not jump Logical flow. Okay. Yeah. Now, uh, yeah Now many people when they're dealing with Matrices, okay They think that the normal laws of matrices And the normal identity laws will work on that. For example, let's say if I say a plus b square Okay, when I ask people, what is a plus b square? Where n, b are matrices. They say sir a square plus two a b plus b square Please note that such laws will not work unless until a b is equal to b a Okay, so unless until The matrix multiplication of a and b are commutative Please do not apply any binomial expansion formula on it Right Are you getting my point? So in general, please do not write a plus b the whole square as a square plus two a b plus b square But if you know, they are Commutative then only you can apply your binomial theorem properties. So this is something which is very important. Please note it down I'll write it over here if a b is equal to b a then Binomial expansions can be applied Can be applied to matrices Okay And especially if you're dealing with a square matrix and one of the other matrix is i Then you can definitely apply binomial expansion on that because a i is equal to i a A being a square matrix We'll take some questions on that in a few minutes time Okay, but let me just complete with this property Then I'll come to that concept Okay, so in general if you say a plus b whole square, how do you write it? This is the way to write it I'm running out of space over here So if you say a plus b square it is basically nothing but doing a plus b Times a plus b. So ideally the result here is a square plus a b plus b a plus b square So unless a b and b are same do not club it Okay So don't start using your normal identities on this. For example, if I say a plus b into a minus b Right, what's your result going to be? It's a square minus a b plus b a minus b square So if a b is not equal to b a don't cancel it out. They may be different different matrices Are you getting my point? So these are some common mistakes which people do Next property Matrix multiplication is distributive over addition or subtraction Multiplication is distributive is distributive over over addition And subtraction Okay, what do I mean by this? So if I do a multiplied with b plus c you can write it as a b plus ac Or if I do a Multiplied with b minus c we can do a b minus ac. Okay, so these rules will distributive rules will work on Of multiplication over addition or subtraction Okay Now the fourth rule is slightly important If you say product of a and b is null Okay, does it mean either one of them has to be null or both have both of them have to be null Yes, so that is something very important. It does not imply That either a or b is null Or both are null Even if two non null matrices multiply they can give you a null as the answer Okay, I'll give an example here. So let's say a is a matrix like this zero minus one zero zero Okay, as you can see it is not a null matrix And let's say there's another matrix b which is one one zero zero And you can see this is also not a null matrix But when you multiply a b you will all check it out that see follow this Symbol zero into one is zero minus one into zero is zero Similarly first row second column will give you a zero. Check it out Second row first column will give you a zero Second row second column will give you a zero. So answer is a null even though both of them are not null Okay, now this has been A very frequent question in competitive exams where they give a scenario like if product of a and b is null Does it mean? You know at least one of them should be null. The answer is no even if both are not null the answer can still be a null matrix Is this fine? Okay, so inspired from this basically We have the cancellation laws. Oh my god. It's running out of space It's running out of space So if I say A b is equal to ac Can I say b is equal to c? Can I say b is equal to c? The answer is no The answer is no here because of just now what we learned. So if product of two matrices is null Even if a is not null it doesn't mean b minus c is null That means it doesn't mean b B may not be equal to c Right may not be equal to c. I'm not denying the fact that it will always be the case. No, it may not be So it doesn't imply that You know either a is null or b is equal to c nothing like this Okay, so basically left cancellation law fails left cancellation fails Okay, same way right cancellation will also fail. So if you say ba is equal to uh, let's say ca It doesn't mean b is equal to c It doesn't mean either a is null or b is equal to c In fact, I should write here. It doesn't mean A is equal to null or b is equal to c Because I've already put a negation sign over it Okay, so be careful about that So right cancellation Fails Right cancellation Fails Is this fine? Okay, before we move on we'll try to take up some questions on this because we have been doing theory for the last one hour or so We'll take some questions. Okay, so we'll take up We'll take up this question Just write it out And let me know if you're done See question is if x is given to be 0 1 0 0 square matrix of order two then prove that p i plus q x is given as Sorry, p i plus q x to the power of m is given as p to the power M i plus m p m minus 1 q x for all p and q belonging to real numbers How do you prove such kind of questions have you done such kind of questions in your practice? Okay, not done. Okay. No worries. What can you try it now? So can you use p mi? Right, there are two ways to do it. One is by the use of p mi other by the use of the properties that we discuss Do you want me to solve it for you or would you like to try with p mi? Okay, try it out with p mi Now multiplication is not a repeated addition anymore. Venkat here multiplication is transforming something you're transforming a linear space by a Transformation agent that is the meaning of matrix multiplication Okay, sir That's why we have to first overcome that feeling of applying normal number, you know rules to matrices Same day we have to overcome putting angles in inverse That's absolutely correct especially you have been making that of that mistake quite a lot now See uh Siddhartha when I wrote a to the power n I very clearly wrote n belongs to i plus correct so See there are some operations which we normally entertain in the matrices for example square root can be found out for a square matrix Okay, inverse can be found out for a square matrix But let's say if you do a complicated power like you know raising it to a power of you know Minus five or to the power of five by six and all such operations are not entertained So we don't we don't raise the matrix to the power of zero Okay So such operations will not be entertained under matrices. We can raise it to either integer power And there's one special symbol, but this is not an operation. It is just a You can say it's a you can just like f inverse. It is a reserved symbol F inverse is meaning that means inverse of a function In the same way when you do a inverse it is basically finding the inverse of that matrix It is not to be treated as doing i by a or something. No, this is not Even though you are your operation wise. It is a inverse is equal to i But we don't reciprocate any matrix like this. There's no operation done like this But the point is that but when we do a pmi We first show that p of one is to write You're then you have to raise Then start okay. Okay, sorry And this m is raised to the power of It is not on any matrix. It's not on any matrix. No, that's correct aditya. Okay, so let's say this statement is This statement is basically pm Okay, pm states that Pi plus qx to the power m Is p to the power mi Plus mp m minus one qx So p1 States that pi plus qx is equal to p to the power one i Plus one into p to the power zero qx, which is nothing but Pi plus qx Which is definitely true Which is definitely true. So therefore p1 is true now Let us consider pn to be true Let pn be true. Okay. That means you are taking This to be for granted. That means this expression is correct I'm sorry, this will be n now what I have to show I have to show my examiner that I have to show that this is how you write it. I have to show that Pi plus qx to the power of n plus one will be equal to p to the power n plus one i Plus n plus one p to the power n qx. This is what you have to show. Okay So now, how do I show this? So we'll assume that this information is correct. So what I'll do is I'll multiply with pi plus qx on both the sides so Force factor multiplication will do Doesn't matter even if you do pre facto with pi Plus qx Okay, so let's see what Transformation happened this into this will be p to the power n plus one. I no problem with that this into this will give you P to the power n qx This into this will give me I square is I only Yeah, so this into which one I was doing this into this. Yeah, that will be n p to the power n Okay qx now xi see i is basically the identity element as I told you so a i and i a both are equal to a only Okay, provided a is a square matrix A is a square matrix So the last term would be if I'm not mistaken It will be n p n minus one q square x square Right now, what is x square if you figure out your x square as aditya rightly said x square would come out to be null check it out So row with column will give you zero this with this will give you a zero This with this will give you a zero this with this will give you a zero. Okay So what will happen this will become a null matrix This will become a null matrix. So basically Null matrix added to any matrix will give you the same result So this will be the only thing that will be left off by the way If you take p to the power n qx common n plus one will come out This is what we wanted to prove She's your RHS Okay, so you can conclude that P n is true implies P n plus one is true and hence By principle of mathematical induction P m will be true For an all m belonging to natural numbers Okay So these kind of questions you can use your PMI Now, let's say I don't want to use PMI and solve it. How would I do that? So I'll tell you another method to do it method number two Copied this one. Can I move on to the next method? Plus Okay Method number two See when you do P I plus qx to the power of m Do you realize that in this case? Let's say I call this as matrix a and I call this as matrix b Do you realize that here commutative property will hold true? Do you agree with me or not? So if you do ab what do you get you get p qx And if you do ba it will become again p qx no difference in the result That means binomial expansions can be easily be applied to it without much problem So if I apply a binomial expansion, it will become m. I hope you remember all your binomial expansion mc1 Pi to the power m minus one qx Plus m c2 Pi to the power m minus two qx square And so on and so forth. Okay Now here you would realize an interesting thing as what aditya had pointed out x square will start becoming Null so any higher power of x will start becoming a null Correct So post this Everything will start becoming null null null. So your answer can only be obtained from the first two expressions Right Is this fine any any questions over here? So this will become p to the power m m Now i to the power m is like multiplying i To itself m number of times which will result into an i Okay, this will become m times p to the power m minus one i Into qx So that is nothing but p to the power mi plus plus m p to the power m minus one q x which is your r h s Is this fine so How do we get from there to where it's very simple you are multiplying pi with pi with pi m number of times right And as I have told you in the scalar multiplication properties, you can actually collect all the scalars at one position Okay, and i multiplied to itself any number of times is going to give you an i only so this term is what is going to appear I hope there's no doubt regarding that Sir, how did you determine a v is equal to b a How do you this is your a and this is your b if you multiply it what do you get you get pqx Okay, yeah, b a will also be pqx Okay, you got it Yeah, I understood sir. Okay Now something very interesting here all of you please pay attention Sometimes we talk about matrix polynomials Matrix polynomials Okay, what is the matrix polynomial concept? Sometimes they will say this is your matrix a Find a q plus five a square plus six a plus seven i Have you come across such questions? Yes, I'll I'll give an example. Let's say the question will speak like this. Um, let's say a is two three five Uh, let's say nine Okay Fine, they'll give you questions like this find a to the power four five a cube minus seven a square plus uh six a plus four i For i2, okay, of course you're dealing with a two by two So these kind of questions are basically categorized as a matrix polynomial question So basically why it is called matrix polynomial because if you see there's a polynomial You can make out of it like this And it appears as if somebody is asking you to find f a Okay, it appears as if somebody is asking you to find f a both of them are same things Okay, so whether you do this or whether you do this it means the same thing. Okay Now here what happens when such a question comes people waste a lot of time Of course doing these kind of multiplication. For example, here you will find a square first correct Because you need a square Then you'll multiply again with an a to get a cube Then probably you'll multiply a square with a square to get a to the power four and then do this, you know collective addition subtraction job That will kill a lot of time Especially in j kind of exam. We don't have so much of liberty to do these, you know These tasks it'll take a lot of time for you. Okay So to overcome this there is a very important method which is beyond your syllabus right now The method is basically called as the kelly-hamilton theorem kelly-hamilton theorem Everybody please listen to this very carefully. It is uh, you know, slightly difficult to understand in the first go Okay, the proof of it. We are not going to talk about it because it is right now not in your scope Okay, so I'm telling you something which is extra Okay So what is this kelly-hamilton theorem kelly-hamilton theorem is a very simple theorem this says Every square matrix Every square matrix Satisfies Its characteristic equation Now, I'm sure you would have not understood. What is characteristic equation? Never mind. That's what I'm going to explain you now What is the characteristic equation? What is this by the way? Okay And what do you mean when you say every square matrix satisfies its characteristic equation now see Characteristic equation. I'm assuming everybody here knows basics of determinants Not a very high ponder determinant basics, you know, everybody Anybody who doesn't know Even abc of determinant I don't say Who is that? Whereas you didn't do cross product perfectors in physics Yeah, so that's determinants. Yeah, that's determinants I'll just I'll just tell you how it works. See when you're evaluating determinant, let's say abcd I'm just giving you a two by two example You just cross multiply this cross multiply this and subtract Okay This is how you evaluate the determinant so determinant Of this matrix gives you a number Like this At least this you know Okay So characteristic equation of a matrix a is basically Determinant of a minus x i equal to zero Okay, so a let's say I take my example here itself as my a two three five nine Okay, so what is a minus x i? So two three five nine If I subtract x times i i is basically one zero zero one By the way, even if I don't mention the order, please It is up to you to take it of the same order as a Okay, then only they will be confirmable to be added or subtracted So this basically if you write it becomes two minus three x I'm so sorry two minus x my bad three Five nine minus x am I correct? Let me know if I missed out anything Is this fine Any problem in understanding a minus x i x is something I'll tell you what is x here So basically to make an equation you need a variable right so that is that variable x Is this clear everybody's happy nobody's saying anything Yes, sir. Okay. Now determinant of this See whatever I'm telling you you will not find it in books. Okay So please listen to this carefully Determinant of this put it to zero that means two minus x Times nine minus x minus 15 equal to zero. Okay If you simplify this what do you get tell me x square minus 11x Plus three plus three Okay, this equation is what I'm calling as the characteristic equation So what this kelly hamilton say these kelly and hamilton were two mathematicians who have done a lot of work on the field of matrices So these guys said that even a will satisfy this equation That means even if you do a square minus 11 a plus three i you will end up getting a null So what do you do replace your x with a and of course three will be replaced with three i Zero will be replaced with none Okay, so this is what these guys told If you don't trust me What are the names again kelly and hamilton kelly hamilton I think they were britishers I'll just show you how it works By the way, you would be now aware that you can apply geo zebra. You can apply matrices Our concepts in geo zebra also. So let's say Yeah, can somebody dictate me a What was a two three five nine Yeah, so this is how you write put a curly bracket Two comma three in one curly bracket then put again a comma then again put a curly bracket five comma nine So this is how it'll come out two three five nine. Okay Now what operation I said a square no So a Into a it'll become a square Minus what did I say next 11a 11a 11 times a Plus three plus three i Three I I let it like oh by the way, you call you've already seen it is minus three i coming up so three i I don't know whether I it has understood. Okay. Anyways I'll define i also one zero Zero one Yeah, so a square a a square minus 11 a plus Three i correct What is this plus 11 or minus 11? minus minus 11 a Plus i Yeah, as you can see it is giving you null correct Isn't it Now if you know this then your life becomes really very easy. So what do we do here? It is like finding the you know Caution in the remainder when this polynomial is divided by the characteristic equation So what do we do in order to find this lengthy expression? The trick is you take the characteristic equation If i'm not wrong, this is the characteristic equation right plus three And you perform a long division with this particular You know polynomial matrix So x to the power four Plus five x cube minus seven x square plus six x plus four Okay, doing this is much more easier and convenient than actually putting those you know finding those matrices separately So we'll do it very quickly see x square. It becomes x four minus 11 x cube Plus three x square subtract it 16 x cube minus 10 x square Plus six x plus four then again you'll have 16 x So 16 x cube Oh my god. This is a very huge expression 11 into six will be 176 176 x square plus 48 x Again, if you subtract you get 166 x square Minus 42 x plus four Then put a 166. So you'll have 166 x square Oh, you very very big figure. What is this 11 into 166? I hope I have not done any calculation mistakes anywhere. Just check 166 1826 plus one sorry minus 1826 Okay, and then this and this will become Okay, so if you subtract it you finally get 17 How much is it? 1784 right? Okay, and this will become minus 494. Okay. So the trick here is that This expression that is a to the power four plus five a cube minus seven a square plus six a plus four i Was basically a square minus 11 a plus three i Times this matrix which is a square plus 16 a plus 166 i Plus a remainder of Plus a remainder of this Correct. So when you're evaluating this this entire thing will go for a toss Because this is annul matrix Correct. So your answer will just become 1784 times a minus 494 i That will become 1784 times. What was the matrix two three five nine? minus 494 One zero zero one Are you getting my point? So if you evaluate this you will be done with your answer Actually, I took a very uh, you know very strange I made it up myself. That's why the answers are quite ugly. Okay But this is how you basically use Kaley-Hamilton theorem to solve Matrix polynomial questions like this Now again not to be used in school Only to be used in competitive exams Any questions here anybody? Yeah, sure Yeah, yeah, I'll give you one more example. In fact, we'll solve a question on this Well, the same thing would be applicable to three by three matrix also, sir. Yes. Yes any square matrix But there's three by three four by four five by five Sir, is it possible to obtain a characteristic equation of higher power than the polynomial equation? I'm sorry Sir, is it possible? Uh, what if you obtain a characteristic equation of higher power than the polynomial equation? If you obtain a higher power from the polynomial equation then see let's say If somebody had asked you a Single degree, okay Then basically you can do this then you have to find a inverse and all because No, I'll tell you some no hidden advantages of this So when you write it like this And let's say a is invertible Okay, let's say a is invertible then you can also write it like this Okay But that process will lead to you finding inverse of that matrix In the reverse case basically we use it to find inverse also So kelly-hamilton theorem not only helps you to find matrix polynomials But also helps us to find the inverse of a matrix I'm sure you would have done inverse of a matrix in the later stage of this chapter So you can use kelly-hamilton to solve the inverse for example, if let's say This can help me to find inverse of this by using the formula 11 i minus a one third of it Okay, then you don't have to find those I know How many methods you have learned to find the inverse of a matrix? Two sir at least two This is the third method But provided it is invertible Okay, so you have to check first for its non-singularity and then You can apply this kelly-hamilton theorem to get the inverse. So to answer your question I think it was Ayush who asked it Vibhav, so Vibhav if you have a lower degree then probably have to work You know this method would not be that greater method Then you probably can use your direct Multiplication and addition because that is easy for you to take You know take care why you want to pick up a hammer to break an egg when you can do it with a spoon Okay, so it is mostly used when you have a you know very heavy degree polynomial In matrices Okay Now a couple of things before I move on Which are very important Number one, I don't know where to write Okay, I'll write it over here So some important properties of characteristic equation Properties of Characteristic equation number one The sum of the roots of the characteristic equation First of all, let me Let me just Name something here The roots of the characteristic equation Are called are called Eigen values Okay, eigen or eigen whatever you want to perform it Call it it is called eigen or eigen values. Okay The main use of it is in formulating eigen vectors Okay, now what is eigen vectors used for? You learn later on it is used for the process of diagonalization of matrices Anyways, that is not going to be a part of your at least j syllabus So i'm not going to talk about it just for your information. You should know it Okay The roots of the characteristic equation are basically called eigen values Now second property is The sum of the roots Let's say let's say my eigen values are lambda 1 lambda 2 Etc to lambda n that depends upon what is the order of a Okay, if a is a square matrix of order n The characteristic equation will also be an order n equation or degree n equation Okay, as you can see here, we had a second degree equation over here because We had a second degree equation over here because a was second degree Okay, if you have a third degree a will become a If a is a if a is order three then characteristic equation will be of degree three. Okay So if let's say you're dealing with A square matrix of order n Then you will get a characteristic equation of degree n which will have let's say these n eigen values The sum of the eigen values Okay, that is nothing but lambda 1 plus lambda 2 till lambda n Will be equal to the trace of that given matrix Now you can check this out from the given workout that we have done So if let's say there were two roots over here lambda 1 and lambda 2 What is the sum of lambda 1 and lambda 2? If let's say these are the roots The root is yeah, some is my 11 11 Now just add that leading diagonal elements of a You'll get 11 Okay, this property is always true Okay Nice Again, don't ask me the proof because this is something which you have to go inside eigen values you know all those Proofs you have to under you know see what did actually kelly-hemington theorem find out found out from that proof It is just a you know working model which I want to explain you so that you can you know make your life easy in your competitive exams third thing that you will figure out is product of the eigen values product of the eigen values That is nothing but lambda 1 lambda 2 till lambda n will be actually the determinant of a okay Now it's not that you cannot find this out If you generally take a matrix a b cd you will come to know that you know At this position normally you will get a plus c to answer your question Venkat So if you perform this operation in general You would realize that see I just do this operation in general a b c d if let's say this was your matrix a Okay And let's say I found out the characteristic equation. I'm just writing down the characteristic equation Without doing any background work characters characteristic equation would be What would be a minus x times d minus x Minus bc equal to zero. This is your characteristic equation if you open this up Don't you see a plus d coming in the coefficient of x? And don't you see ad minus bc coming up here? So this shows that This shows that The leading diagonals will always be the sum of the roots of the characteristic equation And this will be nothing but the determinant of this matrix So this is the trace of the matrix. This is nothing but the determinant Now you can scale this up to three by three four by four and all so that generic proof I have not going to Talk about here Okay Next important property which is most important one if If lambda one lambda two till lambda n are the eigenvalues of a Okay Then if you subject a to any positive integer power even these roots are going to experience the same powers Let me write k I always write n as the power least realizing that n was also the degree. Oh, sorry order of that Matrix, so let me write k Okay, so these will be the eigenvalues of a to the power k. Okay Where k can be Where k can be either a positive integer Or it could be a minus one now minus one means even if you're reciprocating Don't take my word reciprocating to be literal Even if you're inverting that matrix or even if you're taking the inverse of that matrix Then the eigenvalues of the characteristic equation of the inverse of that matrix will be the reciprocals of the eigenvectors of a eigenvalues of a so let's say If you find out the roots here Okay, let's say if you find out the roots here, whatever they may be If you find out the eigenvalues of a inverse You will find that the roots of that characteristic equation would be reciprocals of the roots of this equation That is what the last property says Are you getting my point? So if I square this matrix and try to find its characteristic Equation and from there I try to find out its eigenvalues I can directly say that the eigenvalues would be the square of the corresponding roots of the eigenvalues of a Okay, so read this again if a has lambda 1 lambda 2 till lambda n as the eigenvalues Then a to the power k will have lambda 1 to the power k lambda 2 to the power k till lambda n to the power k as the eigenvalues Provided k is a positive integer or it is also applicable to inverse of a By the way, the eigenvalues have Sorry So do the eigenvalues have to be real or they can be imaginary also They will be real They'll always be real They will be real What type of questions can you get on this of course one involving polynomial In matrices, so I'll try to take a question on this Yeah, these type of questions are very irritating because it makes you find a q base square and do those operations So such questions can be easily dealt with eigen characteristic equation concept so to answer your question Trippan if you're If your entries of the matrices are real eigenvalues values will be real, but if they are complex which we'll see today itself like You can have A matrix with complex numbers as entries of that matrix then it can be non real also any success with this So one minute sir Can I run a poll? Yes Yeah, so those who have answered please press on the poll Let's see Can we close this in another 30 seconds? 13 of you have responded so far on the poll Just a second Okay last 10 seconds everybody please vote At the count of five now I'll stop five four three two one Yo, everyone Nine of you are still not voted. Okay So maximum janta has gone with option b Okay, let's see b is correct or not See quickly make a characteristic equation out of it a minus xi Determinant is zero Now there's a way to write a minus xi also fast only the elements in the diagonal position Subtract X from it like this Okay, rest all the elements you copy as such Okay Let us expand this with respect to the first row So one minus x times This is going to be a x square minus 2x minus one If I'm not mistaken then minus one Asha people are aware of the expansion of a three by three Who doesn't know who doesn't know how to expand three by three everybody knows it right? I don't See what do we do is I'll tell you how to expand with respect to one of the rows. Let's say row number one Just take this element Okay, write it down over here. Okay Then hide the row and the column to which that falls So you'll see these four elements left out, right? So just put your finger on the first column and first row You'll see these four elements right then cross multiply and subtract it Okay Then next element right one but put a negative sign in front of it So it basically alternates plus minus plus minus plus like that Okay, so write a minus one hide the column in the row to which it belongs Now cross multiply these two which is minus x cross multiply these two which is two and subtract it Okay, go ahead sir. Thanks. So this is done now zero is not to be taken into account So that will be zero now determinant expansion is only done with respect to any one row or any one column Okay of your choice. You could choose any row or any column of your choice Okay, so if you expand this what do we get? Let me be Slightly quick here. Okay Some things may cancel out for example X cube will feature in Okay, minus x cube will feature in I think x square and x square will be three x square So let me just strike off whatever I'm taking care of. Okay, so that I have a track of it. So minus x And plus one Okay, have I have I done it correctly just verify Plus I'm so sorry. This is plus x This and this will Wait, wait, wait one second. Uh, this will become uh plus x plus two correct So this will become x square minus x minus one minus x cube Plus two x square plus x that's correct Okay, oh x x it will cancel out. There'll be no x only My bad. My bad old age problems. Okay So now this will become now. This is the characteristic equation now Matrix A will satisfy this characteristic equation. So you'll say minus a cube Plus three a square plus two i will be null I think if you can just take the negative sign everywhere Oh my bad. There's a minus one over here Plus one Okay, so now this is going to be your final result. I think option b is what is matching with it, right? By the way, this is not zero It's actually null There is null actually, okay, it is not a it's not a zero Matrices will give you null matrices as the answer So option number b is the right option Okay Do you find that this method is more convenient than actually finding a cube? an a square and all those stuff Yes, yes But unfortunately not to be used in school. Sorry Okay Let's take another question. Actually, there are so many questions. I would like to do with you, but I Have to pick them out. Just bear with me This is one we did ai, sorry itj 2005 a is given i3 is given. I don't know why they have given i3 even if they have not given we would have figured it out And they're saying a inverse is this find cnd values Let me put the poll on once again. Okay They're not even to see the poll Is it I minimized it Yeah, after you minimize it, I don't see anything What's done in the taskbar Oh, yeah One person has answered after two minutes of hard work Why you want to go to that extent aditya? No co-factor in all is required See if you go by that co-factor root, basically you're finding a joint Then you have to find determinant. Then you have to find a square. You have to do this operation compare the Terms. Oh my goodness If you're going to solve it in that way, I better Quit this question and go to the next one Jay is not testing your uh conventional thinking approach Convention thinking anybody can solve it if I give 10 minutes to somebody he will easily be able to solve this question Okay in another 30 seconds, I'm going to close the poll another 15 seconds five seconds five four three two one Go Half the people have only voted come on class come on Okay, yeah, so Janta is saying c option. Okay Let's see whether Janta is janadan or not See character seek equation quickly Let's not waste time and you know the stuff you should not waste time Okay, expand with respect to the first row so one minus x one minus x four minus x Plus two nothing else you have to do. This is your characteristic equation. Correct. Of course, you have to expand it once more If I'm not mistaken will be x square uh minus five x Uh plus four plus two is plus six Okay, keep correcting me if I miss out something Okay, so now if I expand it, uh, if I'm not mistaken I'll get x square minus five x plus six minus x cube plus five x square minus six x is equal to zero minus x cube term x square term will be six x square gone gone minus eleven x gone gone Six equal to zero. Okay. That means your characteristic equation is a cube minus six a square plus eleven a Plus six i equal to o What the point Now as I told you in while I was dealing with characteristic equation characteristic equation not only helps you to you know get the polynomial matrix polynomial question See I told you So multiply with a inverse now provided a inverse exists. Now, how do you know whether a inverse exists? If the determinant here is non-zero A inverse will exist as you can see determinant here would be four plus two six non-zero. Okay So a inverse exists. So for that multiply throughout with a inverse. That means reduce one one power each This guy will become 11 i this guy will become minus six a inverse Take it to the other side a inverse will be one sixth of Correct me if I'm wrong That means your answer is C is six minus six my bad D is 11 So minus six comma 11 is your answer option number C Can you beat this method? I don't think so. Any other method will take you double the time at least Okay So say thanks to Kayleigh and Hamilton Me also no problem Okay, let's take a break now on the other side of the break. We'll talk about You know transpose of a matrix and other special matrices like an idempotent involuntary Orthogonal all those stuff Let's have a break Is this fine enough so let's have a break Yes, sir question Yes, sir, could you go back? Are some some people are copying this right? Yeah, that's what I meant Go back wins go to the previous slide Oh, yes, sir People who are copying this One second, sir Now this method you will not find it in books also. So please don't expect this to read out from some books These all are engineering stuffs So I once upon happened to do engineering. So there I learned this Yeah, thank it This is this a page. Uh, no, sir. The previous and where you discuss is Kayleigh Kayleigh Yes, sir. Kayleigh sir This one Ah, yes, sir Left down Yeah, yes, sir Sir most exactly what you do after finding out the characters Like that polynomial relation, I didn't understand this this you got Oh, yeah, I understood that place your x with a and a constant with III constant I that's what we do So now this is always, you know null, no So anything multiplied will also be null correct Okay, and from here also you can multiply with a inverse also, no So a inverse will give you a A inverse a will become I I a inverse will become a inverse so I can find a inverse also from this Oh, okay Not this one the actual polynomial thing you divided it or no Basically, I I treated I treated it as a polynomial. So basically, how do you write Dividend as quotient times divisor plus remainder Oh, okay, right this I did the same thing. So I treated this as the dividend Okay, this is my divisor This is my quotient And this is my remainder Yes, sir. So since divisor itself is zero or sorry null this whole thing will be null, no Yeah And only this thing has to be found out. So for finding this big thing you can just evaluate this thing Oh You can just deal with a and I only that's it Oh, okay. Yes, sir So and you obtain that by quadratic on your own or so just multiply two random things to give that two factors No, this is a long division method No, no the actual This I made up on my own. Yeah, sorry. I made up this on my own I am So, dude, when do we have break? Once again break has been declared. Okay. Once again. So six 29 we can meet Is this fine? So now on the operations, uh, basically we have learned addition. We have learned A scalar multiplication. We have learned how to multiply matrices And we have also learned how to, you know, find answers to matrix polynomials by the use of kailey-amilton theorem Now we have been equipped with enough knowledge now to now talk about, uh, you know, special types of matrices Uh, but before that, I would just like to talk a bit about transpose of a matrix transpose of a matrix So let's say, uh, there's a matrix a which is basically, uh m cross n order matrix, right when we say transpose of this matrix. Basically, what do we do? By the way, representation is a to the power t in fact a superscript t I'll not say to the power or a dash many people write it. It is basically switching the position of It is switching the position of the rows and the columns or swapping the positions of the rows and the columns. Okay So when you do transpose, what do you do? You send the you send the element in the i jth position to i j ith position Okay, and as a result, you'll see the order will become from m cross n to n cross m A simple example for the same is let's say we have a matrix a which is one two three four five six Okay, then it's transpose would be written as One two three four five six So you can see a two by three matrix has now become a three by two Now let's say this element if I talk about it was present in the first row's third column So this is basically a one three element first or third column It has now come to a three one position. So that is what is meant by this Okay Now some properties of transpose because these properties are going to be very important later on So everybody, please pay attention to these properties Properties of transpose of a matrix transpose of a matrix first property is If you transpose a matrix Twice it'll give you the same matrix back, which is very evident So a transpose transpose will give you the same matrix back Okay If you transpose the sum or difference of two matrices It is as good as taking the sum or difference of their transposes Okay, if you transpose product of two matrices It follows Law of reversal Okay, this is called the law of reversal Okay, so this was I think the question that somebody was asking me early in the class today Okay, and this can be extended to any number of you know matrices. Let's say a b c transpose Then basically you will transpose it from the reverse direction. So it'll be c transpose b transpose a transpose Is this fine? Okay This is one of the most important properties that we'll be exploiting For you know subsequent concepts coming up in matrices Next property is Next property is if a scalar Multiplication of a matrix is transpose. There is no effect on the scalar term. Okay. Don't start transposing a scalar number There is nothing called transpose of a scalar number. Okay, so k a transpose is k times a transpose Similarly, if you are raising any square matrix ism is a square matrix Let's say is a square matrix And if you're using it to any positive integer power and you're taking a transpose of it. It is as good as Raising the transpose of that matrix to the same positive integer power This is also applicable to Also applicable to inverse also later on we'll learn about this So a inverse transpose is a transpose inverse. There's no difference in this But many don't treat inverse as a power. Okay. This is not a power. This is just a symbol It is just a symbolic thing a inverse means inverse of the matrix a Next property is There's no change in the trace. There's no change in the trace even if you transpose the matrix Okay And you learn later on also there's no change in the determinant value also Okay, but I'll not write it down over here. We'll talk about it when we do a determinant properties trace of A product of a matrix with its transpose Okay Of course a a should be a square matrix over here Okay, this will always be greater than equal to zero By the way a into a transpose will always result into a square matrix So this will by default be a square matrix Why it happens to be greater than equal to zero because the elements in the leading diagonal will all be perfect squares Okay, now With this knowledge, we're now going to jump to some special matrices There you would realize how important these properties are Okay Yeah Yeah, sure done people who are copying are you done with it? Yeah, okay So i'm not going to the next page Here we'll talk about Some special matrices Some questions have been asked in je on these special matrices. Okay, so let's be aware of them The first i'm going to start with is idempotent matrix What are idempotent matrices? So first of all idempotent matrices are square matrices. So these are square matrices Okay, this satisfies This satisfies this property that a square will be equal to a Okay, now this is the fundamental property, but you realize that if you raise it to any power Okay, that will actually giving you a only for example if I do a cube Okay, won't it be a square into a and a square is again an a so a into a is again a square, which is back to a right? Correct So if I raise it to a power four that means basically you're doing a square square That is again a square that is again a correct So the idea is idempotent matrix even even they say a square is a it is basically a raise to the any power Of course n should be positive integer That will be a itself. Okay An example of the same would be let's say I take an example let's say Two two minus one minus one Okay, if you do a square You can just do this operation very easily at your end also So take this row multiply with this it'll become Two only this with this will become again a two this with this will become uh minus one And this will this will become again a minus one. So you're back to a only Okay, so this is a typical example of an idempotent matrix Now few properties that we need to know about idempotent matrix Uh, remember, uh, ncrt doesn't ncrt or your basic, uh, you know books of your school level They don't cater to these special matrices do they? Okay Sorry Not there, no Okay Now this property is basically Treat them as a prove that question. Okay If a and b are idempotent matrices If a and b are idempotent matrices Okay then ab is idempotent If and only if ab is equal to ba Can you prove it Only when a and b is Multiplication is commutative then you can say That ab will be idempotent provided a and b themselves are idempotent Let me know once you're done. See a and b are idempotent means what a square is a b square is b Okay, primarily, this is what we mean by a and b mean idempotent So if you want ab to be idempotent What does it mean? It means you're saying ab square should be equal to ab Now what is the meaning of ab square ab square means ab times ab Okay, so we have to prove this Okay, this is what we have to prove So now let us start with the lhs Lhs is ab whole square ab whole square is ab into ab correct So this is nothing but a ba into b Correct Now if you want your result To come out to be ab and you have to use this property, then only this will become ab again Then only it will become a square b square And then again from these two properties you can say this is a and b So unless until this is true, you will be stuck at this step So because this was true, then only you could proceed further and write it like this and then this and then this finally So if you want ab to be idempotent It can only happen when you break this bottleneck Or you break this hindrance point over here That is you consider your ba and ab to be the same That means you honor this condition Then only you can proceed further and get a ab Are you getting my point? Next property Can I One second. I'll just copy this down. Yeah Sir, can you extend my ab will be equal to ba No, I'm not saying why ab will be equal. I'm saying that if this condition is not true No, then it is not going to be idempotent Only when this condition supports it, then only ab will be an idempotent matrix See, what does the property say? Property says if a and b are idempotent Then ab will be idempotent if and only if ab is equal to ba Got it So what I'm doing here is I'm basically Starting from the fact that if ab is an idempotent matrix and If this condition is not True, then you will be stuck at this step You will you can't move any further So since ab and ba are equal Without that it will not be idempotent So I have considered ab as ba or in fact ba as ab and then I proceeded to get my final result Okay So the question in the exam can be placed like this If a and b are idempotent And ab is also idempotent Then which of the following option is true in that one of the option will be ab is equal to ba That option you have to mark getting my point now second property If a and b are idempotent If a and b are idempotent Okay Then a plus b will be idempotent if and only if ab is equal to ba is equal to a null matrix Now this is very easy to prove So if a and b are idempotent means you're saying a square is a and b square is b Okay Secondly if you're saying a plus b is idempotent that means You are claiming this Okay, let's see what happens when you go next So when you do this it becomes a square plus ba plus ab plus b square on the left hand side Sorry b square, okay on the right hand side you are saying a plus b What does it mean? It means ba plus ab Is going to be Now cancellation law apply cancellation law so null will be left Yes or no So can I say when this condition is true Then both of them will be null null But why both of them have to be null? So if one is not null then obviously a a plus b whole square won't be a plus b Sorry, I didn't get you So if ba is null and ab is not null Then you have a plus b whole square there's a square plus b square plus ab Okay So we'll still have an ab term So That was not very clear Why I mean there's some could be null why individually they have to be null that is my question If one is not null and the other is null Then how can there be idempotent in the first place? I didn't get you It's like saying a non-zero number equals zero if you want if that means idempotent thingy won't be a satisfied here Athar was an answer No, no, Athar my question is I could have the sum as null but not individually. So let's say this was a matrix x and this could have been a minus x Okay, yx has to be null null each that is what i'm asking you And odd power thing is also there Sorry Odd power What odd power is my is my question clear to all of you They both are not null then they can't be negative of each other I think Like ab can't be equal to minus ba I don't know how to prove it but that may be the answer Okay, think about it and let me know Some of you have given the right argument you just have to Properly phrase your sentences Okay, next question is Our next property so as to say is If ab is equal to a And ba is equal to b Okay, then a and b are idempotent this question can Come in in your combative exams. Sorry idempotent. Can you prove this? If ab is equal to a And ba is equal to b it means a and b are idempotent So using this you have to prove that a square will be a And b square will be b Isn't this very simple if you do if you take ab is equal to a Yeah, and multiply with a ab. Let's say Okay So can I say Yeah, can I say here this ab could be written as a Correct Sorry this ab could be written as a and a into a could be written as a square And this ab itself is a I'll repeat once again See Let me start with We know ab is equal to a We know ab is equal to a Correct If you multiply with a a post facto this will become a square Correct Yes or no And this ba You can write it with a b And ab is again an a so you can write this as this that means you prove this statement Right, so a has to be idempotent Similarly if you start with a ba is equal to b Okay, multiply with a b Then this ab could be written as an a And this ba is again a b Which means b is an idempotent matrix So so a is equal to b Huh Yeah, when we do that it'll become a is equal to b because in that first case if we multiply by b Okay, then we'll get ba as b Which one which one you are saying the first part this part Yes, okay, so ab if you're multiplying with a b Huh, what will happen? Yeah, then you take ba as You multiply pre facto or post facto Uh pre facto Means b here Yeah, okay. Okay, okay fine. Uh, yeah So a a need not to be equal to b right or that's ana sana Or an ananya ananya ananya, so a need not be equal to b. Why should ab equal to b? Mm-hmm. Yeah Okay, so next property is If a is idempotent If a is idempotent Okay Determinant of a could either be zero or one now This is something which would require you to know your determinants properties also So I don't know how many of you know this property if a and b are square matrices If a and b are square matrices Okay, then determinant of ab is determinant of a into determinant of b. Are you aware of this property? This is a very very important property in determinants Okay, so if you're saying a square is a Right, basically you're trying to say this So basically you're trying to say This so from this property you can say this is determinant a into determinant a is equal to determinant a Right, that means determinant a square is equal to determinant a Okay, so this can imply two things either determinant is one or it is zero Okay, so for any idempotent matrix Your determinant will either be zero or one. I think you can test the example which I have given you as you can see Here the determinant is going to be zero Any questions here anybody? Down yeah, this is the lowest I can go Yes, just a second Next we'll come back to problems on these once we have completed all the special matrices Because normally they all come together in a problem Next is involutary matrix The word itself means self inverse Right, so a square matrix A square matrix a is said to be involutary if A is its own inverse in other words A square is i Okay, so two things that we need to take care That it should be a square matrix. In fact all the special matrices that we are going to talk about there will be square matrices only And second thing is its square should be identity matrix Now you would realize here that from this property it is evident that when your a is raised to any even power Okay, it will always give you an identity matrix Okay, and when a is raised to an odd power, it will always give you an a I hope it is very clear from this property Correct, so if you see a cube it will be this into a so that will generate an a But if you do a for it will again become an i So in an involutary matrix whenever you raise the matrix to a even Index or even power it will give you i but if you raise it to an odd index it will give you A itself An example of this can be Let's say minus five minus eight three five Okay, just try squaring it Just try squaring it. So this will be 25 minus 24 one This will be minus 40 plus 40 zero This will be 15 minus 15 zero. This will be 24 plus 20 minus 24 plus 25 one. Okay One important thing that you need to note here is determinant of Involutary matrix will either be one or a minus one It again comes from the same fact. So from here if you do determinant a square It will be determinant of i determinant of i is one and this is determinant a square So this will give you two results plus or minus one You can check here in this case the result is Minus one right check the determinant here. It will be minus one Other way round is not true. Okay, so don't think like our determinant is one So it has to be an involutary matrix. No other way round is not true If it is an involutary matrix the determinant will be plus minus one same with the you know Item potent also if it is an item potent matrix, then the determinant will be zero or one But vice versa is not true Is this fine any questions here Uh, sir, I had it out in the previous slide actually previous slide in the Item potent. Yes Yeah, tell me sir in those cases where we were multiplying by a or multiplying by b, uh, if a or b were Null null matrices, then we couldn't have done that A and b were If either of a or when we were multiplying by a or multiplying by b if either a or b were null matrices, then we couldn't have done that No, but see in the item potent matrix we normally None matrix are by default the item potent matrix that is agreed. No problem with that But why why won't we do that in the case of a null in this case? I didn't get is null an exception sir. It's item potent Null is an item potent that is fine. What is item potent square of it will be the same matrix, right? Yes, sir So why won't it work in this case? We can't multiply Sir when we multiply by null then it's like we're multiplying both sides by zero But this property is getting true, right? So a will be null. So null is equal to null square. That is correct. No Yes, right So that is a special case, but it still satisfies it It is no no exception to this case, right? Yes, correct. No, yeah so, uh Yeah, any other question Yes, sir. Yeah, sir. Can matrix be an important and involuntary? Can a matrix be item potent and involuntary? Uh, let me check I I is such a matrix, right Okay, sir. Yeah, because I I square is I and it is The same matrix itself, right? So yeah, it can be involuntary and I am putting it in the same way. Yeah Okay, sir Next is Null potent Null potent So this is also a square matrix This is also a square matrix Let's say a So when a matrix basically gives you this property a to the power m is null But a to the power m minus one is not a null Then we say it is a nil potent matrix of index This is called nil potent matrix of index m matrix of index m For example, let's say if you take a matrix, okay if you square it For let's say it's a not a null matrix if you square it Okay, it doesn't give you null if you cube it. It doesn't give you null But if you raise it to the power four it starts giving you null Then we'll then we'll say that this is a nil potent matrix. Then a is a nil potent matrix of index four Okay, nil potent matrix Of index four Let's say there's a matrix a which is a square matrix, which is not a null matrix Okay, if you square it it gave you a null non-null if you cube it it gave you a non-null if you raise it to a power four Then started giving you null In that case a will be a nil potent matrix of index four Okay, right Uh, I'll give you an example on this. Let's say, uh, if you have a matrix like this, uh, a b b square minus a square minus a b Okay, right now. Let's say it is not a null matrix. Let's say a and b are non-zero quantities Okay, try doing a square What do you get a square is basically This into this Okay, so a b square minus a b square zero a b cube minus a b cube zero Then minus a cube b minus plus a cube b zero And minus a square b square plus a square b square zero, right? So basically we'll say that this is a nil potent matrix of index two Matrix of index two So the least positive Power for which it starts giving you null that will be called as the index Now note that after a to the power four in the previous in an example hypothetical example, which I took It'll be zero zero zero or null null null, right? So the least power for which it starts giving you null that is called the index of that nil potent matrix So can it have a not null value after a few indexes? No, no, no, it won't moment one null has come. It'll be null only Any square matrix will be null only, correct? Yes. Okay So one second could you show the example? Yeah So basically if a is a nil potent of index m Nil potent of index m then A to the power n will be null If n is greater than equal to m This is to answer your question Anjali So any higher order higher index if you take I'll give you null only Yes Okay, another important thing is Can you comment anything about the Can you comment anything about the determinant of a nil potent matrix? Zero Will it always be zero? I'm not able to exactly does this nil potent matrix work, right? I told you this worked in this way that after a certain value which is called the index It's the terminate start. It's a value starts becoming null Yes, sir Like what why does it work the way it does like There's some graph or something for this See It says that if you apply that transfer transformation repeatedly Okay, then that transport that transformation will collapse the r2 space or whatever space it is working on To a point toward, you know null space Okay, this requires your understanding of linear algebra a bit See when you apply when you are multiplying a matrix with itself Basically, you are imposing one transformation on the other see Remember in the beginning of the class I gave you Oh, yeah increase in dimension Right, if let's say if I apply one more multiplication on this let's say one Let's say I did one two three four What does it mean? It means There are two transformations which are working together So this will be an A combined transformation. You can say let's say t Are you getting my point So it is like let's say I give an example that you rip. I showed that my daughter's image, right? And so I did it about the x-axis Let's say I then reflected about the y-axis then I asked you tell me a transformation which will directly result into that Then that would be the product of those two transformations Are you getting my point? Yes, sir Get the get the motive behind why we are learning these matrices matrix multiplication is basically Trying to do multiple transformations and see what transformation will equivalently replace that transformation right Let's take this. Let's take the same situation. Let's say this was my daughter's photograph Okay, if you want to reflect it about the y-axis what transformation we used Replace x with minus x. So what what did we do that minus one zero zero one? Yes, sir. Let's say whatever image comes here. I want to reflect it again about the y x-axis. What transformation will you use? Minus one zero zero minus one You'll use one zero zero minus one correct from the first thing. Okay, okay Now the transformation which directly brings from this position to this position will be nothing but t2 minus one T2 t1. So just multiply these two. See what will happen? So one zero zero minus one multiplied to Minus one zero zero one. See what will happen? It will become a minus one check This will become a zero This will again become a zero and this will become a minus one. See what has happened If any pixel has a position vector x y it will result into minus x minus y Showing that it is jumping straight away from the first quadrant to the third and vice versa. We have to multiply them in the opposite order of Yes So if you're doing transformation t1 then t2 then t3 then t4 then the multiplication will be t4 t3 t2 t1 like that Yeah, yeah So try to understand matrix multiplication is basically applying multiple transformations and trying to produce a single transformation which produces the same act Now are you getting the point? So how do you how do you know express this in terms of a graph or something? I have no clue about that, but this is how I understood my matrices I don't know how you brought a graph of a matrix. I don't know that They are just agents to make transformations of space Yes, sir. Okay. Yeah, any other question anybody has? Okay, so think about it and let me know Think about it and let me know. What can you comment about the determinant of an ill-portant matrix? So, yeah I got the answer to the question you asked before about Eigenportant matrix What was the question I asked? So you asked why ab and ba should individually? So, sir, I think if So you get a plus b whole square will be a square plus b square plus av plus ba So if you multiply i to ab plus ba whole It's going to remain the same And i is a into a inverse So you will get a ba plus ba square So could you write the answer? Yeah, sure sure Yeah, my question was If a plus b whole square is equal to a plus b Okay, then we got from here a square plus b square plus ab plus ba, right? Yeah, yes, sir. Okay. Yes. So if you combine a b plus ba and multiply it with i Okay, so now this becomes a plus b Okay, and now this should be a null matrix, correct? Yes, sir. So it's like a plus ab plus ba whole multiplied with i Do I write this or should I not write this? So it doesn't matter. Okay. I'll not write this. Okay, then. Yeah, so i is a into a inverse Okay So if you multiply a with um, I think let's give it a property then you'll get um a square b plus a ba A inverse will be outside A inverse is outside. You're introducing a inside. Yeah So that means you should have multiplied with A inverse Yes, I took it to the right. Okay. So if you write a inverse a and if you open the bracket it becomes a square b Plus a inverse a inverse will be outside Okay, okay. Let me not introduce a inverse Then what next? Yeah So we'll get if we take out um Yeah, um If we take out a I mean one second, sir No problem. I mean I got it. I don't know why one second Okay, we'll we'll come back to it. Uh, Anjali just collect your thoughts. Yeah. Yes So a square is a A square is a fine. Yeah, so you will get a b plus a ba which is a inverse a i plus a If you multiply it again, that will be b. No, sir. Take a b outside now Okay, if I take a b outside it will become A inverse a b It'll become this If you write a inverse a then we can say that a b should be zero No, no, no A inverse a is reduced to an i no Yes, sir. But if you don't write it like that then we must say that Um, a b should be zeros. Um, if the whole term should be zero Why I can't comment. I can't comment on the The two determinants two matrices which are multiplying to give you null that any one of them will be null No, I can't say that it violates the very basic property that we did if a b is null We cannot say a is null or b is null Or both are null correct So I'm like I don't understand like why can't we say that see anyway a inverse is not null And b plus a is not null if we have to prove this So a b should be null. It doesn't matter any if any one of them is not null. We can still get a null That is the whole point. No If a is not null b is not null still we can get the answer as null Okay, sorry, sir. Yeah, yeah Don't treat them like numbers. The problem is most of us we treat matrices like our numbers Okay Since x x into y is zero means one of them have to be zero. It is not true for matrices. It doesn't work for me Excuse me, sir. My power actually got cut. Do you mind repeating what you just said? uh I mean my discussion with Anjali or just in general Oh, I I got cut when you were discussing it with Anjali. Did she get it right? No, no, that was not correct argument, but she tried hard to do it. Well done. Okay So, yeah Sir, I don't Sir, uh, you said that matrices are like transformation. So what is a null matrix to It annihilates it completely. It makes it it makes that space reduced to a point. It makes that Yeah, it makes a vector reduced to a zero vector. Simple as that Okay, sir. So if we apply the null potential matrix 10 times then if you get Yeah, it'll just disappear. Yes. Yes. So basically if you apply that transformation multiple times Then what will happen the combined effect of all those transformations will result into a Such a matrix which will make the entire vector space or 2d vector space or 3d vector space That depends upon in what space you are applying that transformation that will reduce to a point Zero okay Okay, so now we are moving towards orthogonal matrix orthogonal matrix What is an orthogonal matrix again it is a square matrix Okay Such that a into a transpose is a transpose a Is equal to an identity matrix of the same order So any matrix which Performs this activity a into a transpose is a transpose a equal to i It will basically be a orthogonal matrix Okay Now orthogonal matrix is very close to my heart. I remember my very first project in the field of wireless communication was on cancellation of noises in cdma 2000 technology So we used to imply orthogonal matrix right to cancel out Any kind of you know noises which used to be carried along with the modulated signal So it's very very useful in wireless technology also I'm just giving you an engineering, you know Application for it. Well, you may not take these matrices very seriously in class 11th or 12 But later on when that becomes your profession, then you'll start respecting these same matrices I'm sure people in the field of digital image processing would now be like, you know Doing arty and all for matrices. They're so important So, uh Yeah, good old days for me Let me give an example of an orthogonal matrix. So something like this Uh, let me just make up an example. Um, one by root two Minus one by root two One by root two one by root two. Okay. This is basically an orthogonal matrix of order two You can try multiplying it with its transpose So the transpose here would be Swapping the Rows and the columns Okay, so when you multiply you can see The first row with the first column will give you half plus half, which is one The first row with the second column will give you a zero completely Second row with first column Will give you again a zero second row with second column will give you a one So that gives you an i which is nothing but This property. Okay. So this is an orthogonal matrix Sure, Anayita. Yeah Sir, uh, do we just give an example of how to find transpose for a three into three matrix, sir Transpose of a three into three just change the rows to columns. Okay. So let's say one two three four five six seven eight time So if the transpose of this is required, you just write one two three four five six seven eight time Now what are the properties of orthogonal matrix? Sir, should a a into a transpose Equal to a transpose into a should is there a necessary condition or is it fine? No, it comes automatically It comes automatically Okay Sir, can I say that a transpose is also a inverse? Yes, so from this we can also imply that a transpose is the inverse of that matrix. Yeah, obviously properties of orthogonal matrix And we'll prove them as we go on number one If a is orthogonal So will be so will be its transpose Can you prove this? Now this is very easy to prove If a is orthogonal It comes from the definition only Yeah, so basically if you're saying this is true, right Then a transpose into transpose transpose will be nothing but a transpose into a Which basically is governed by the fact that a into a transpose is a transpose into a and hence even is transpose will be orthogonal in nature second property If a and b are orthogonal If a and b are orthogonal matrices So will be a b Can you prove this? See if a and b are orthogonal That means a into a transpose And b into b transpose is i i'm not writing the other part of it And if you have to prove a b is Orthogonal that means you have to prove a b a b transpose Is going to be i This is what you're doing Okay Now if you open the transpose bracket, we all know the law of reversal. So it'll be b transpose a correct Sorry, b transpose a transpose Now this is going to be i as per this condition So it'll be a i a transpose And you know that i Doesn't make a difference. It just becomes a transpose and from this condition. We can say it is back to i Which is your r h s Right. So if a and b are orthogonal, then a b will also also be orthogonal Okay Same will be true for b a as well Now when such questions comes in the accommodative exams, I've seen some people They'll act very strange. They will take an example of two orthogonal methods and try to convince themselves that it works Instead, if you know these properties of transpose very well, you won't have to do it Take an example and try to see which of the option is correct That's why knowing the properties is very important Lastly, even though I have not talked about inverse of the matrix so far, let me tell you if a is orthogonal, so will be its inverse Okay, by the way, the only property that is required here is Let's say if I tell you this property that a b inverse is equal to b inverse a inverse That means it also follows the law of reversal. Can you prove this property? I think so you will not even require this one also But we have already mentioned the Transverse properties So if a is orthogonal You know that this is correct. Okay, and if you want to prove that A inverse is orthogonal. That means you have to prove somehow That this is I correct Now here what we can do is we can write it as a inverse A transpose inverse because as we all know that transpose and inverse positions can be strapped Can I say this to be A transpose a whole inverse basically I'm using the Law of reversal in the opposite direction Can I say this Okay, and since a is an orthogonal matrix, we know a transpose is i so basically it is i inverse i inverse is i itself i doesn't i is not affected even if you invert it Is this fine Anyways these properties I will repeat once again when I'm doing inverse of a matrix with you So how do you get a inverse and a transpose inverse as A inverse transpose This is a property. You know check the transpose property Should I go back to this Yes I can just check on your notebook for you. It'll be just turn a couple of pages and you'll get it Yes Did you find it Yes Okay Now we'll talk about questions based on them first and then we'll move on to The main ones symmetric and skew symmetric which is going to be asked in your school as well Let's take some questions first on them. Okay, so this is a column match question Let us first match the a of column one To column two Uh in your answer, please write it like this a Map to which of them p q r s or if multiple of them is matched. Let me know So let's say he's mapping to p and r. So like like that a map to p and r Okay, aditya has given one response No, sir Yes, tell me when good Oh, you can hear me. Okay, sir. Could you go back to the previous page After this question of course After this question, okay fine Okay, so most of you have replied let's discuss the first one If a is idempotent idempotent means a square in fact higher powers will become all a Correct a q will also become an a and so on Now i and a are commutative right so we can apply binomial expansion on this So i minus a to the power of n will be i to the power n which is i minus nc one a Plus nc to a square minus nc three a cube and so on Till i reach n c n That depends upon n what sign i'll have over here. So i'll write it like this minus one to the power n n c n a to the power n, okay Now i can see that I can see that All of these terms will be a only so basically you are taking a and you are summing up nc one nc two All the way till n c n fine What is this value? What is this value? We all know Yeah, we all know that nc zero minus nc One plus nc two Minus nc three till wherever it goes. Let's say minus one to the power nc n that is equal to zero, right? Correct So can I say if I take all these terms to the right hand side? I'll end up getting this and which is nc zero and nc zero is one. So answer will be i minus a So a will map to s A will map to s I think most of you got that Even like a normal definition itself. That's the thing no sir Okay Oh, yeah, and and yes if you do if you do just do this you will get an idea of the pattern if you do a square You will end up getting i minus a into i minus a That is nothing but i minus a minus a minus a square right Sorry plus a square and this basically gives you i minus a again Correct. So if you start going into a recursive relation, let's say if you do i minus a cube Then it'll be just i minus a times i minus a which is back to i minus a so the answer Any ways come down to i minus a which is going to be an option is Is that what you're saying? Yes Uh, sir That's the idempotent thingy. Um, does that mean Something minus a also to define and also is equal to just something minus a Not in this case. It may not be i for example So you're saying b minus a square will be b minus a if a is idempotent Yeah No, no, no Okay, we can't say so But if both v and a are idempotent then Then you can verify, you know, so in this case if both and and it should be commutative also Yes, correct So if you're saying b Minus a square and b and a are idempotent. So what do you end up getting you'll end up getting minus b square b Minus b a What do you get? Yes, sir. So if this is b And this is a No, I I really doubt that information No How will you generate a b minus a over a? Yes, yes, okay. So You can figure it out on your own Sorry, uh, the question was We know However, you can say if a is idempotent i minus a is definitely idempotent Yes, sir. Okay So A lot of people have responded Can we discuss now? Yes, sir Okay So I A is involutary involutary means it is its own inverse. So a square is i and so on and so forth. So Okay, so can I say this is going to be i to the power n same expression over here. Okay So let me not waste time writing it So in this case, what will happen a square will become i A q will again become an a and so on and so forth, right So you'll end up seeing something like this I Minus nc1 a Plus again an i Again an a and so on That means I will have one plus nc2 nc4 and so on and then minus a will have again nc1 nc3 and so on Correct. Can I say each one of them was two to the power n minus one Okay At least during the time when we did binomial And I'm sure they will continue to be the same now also. So your answer will be p So b will map to p Okay, Venkat c basically says a is nil potent of index two That means two n onwards it will become So this is very simple. So basically from here onwards, you start getting a null. So it's i minus na So c will map to q c will map to q Okay, that means d has to map to r Anyways, we'll verify it If a is orthogonal Then a transpose inverse What will it be? I'm sorry. It'll be a because a inverse is equal to a transpose Okay, so basically It comes from the definition of Orthogonal that a transpose is a inverse Correct. So basically you're writing a inverse inverse Won't it be a again? If you invert an element twice, it'll give you the same element back. So d will map to r Any questions here? Let's move on to the next question. Okay. Now Venkat you wanted me to go to the previous slide Oh, yes, sir Yeah, tell me Take Let me put the poll button on Only one person has responded so far See the pattern in this case You can easily answer by seeing the pattern Okay, last 20 seconds, then I'll close the poll All right, let's discuss this. I think enough time has been given for this. So This is the result for the poll 41 percent. That is the maximum has voted for c Okay, let's see if c is correct or not See it's just a pattern observation the moment we do a cube Okay, we realize we are doing a into 2a minus i which is nothing but 2a square minus a And a square is again the same term. So you may write it like this That actually becomes 3a minus 2i Okay Now from here onwards we can start ruling out the options because when your n was When n is your three Neither this gives you the answer because this will give you 4a Okay, not as this give the answer nor this gives the answer only c is the right option in this case Okay But if you want you can convince yourself even further by doing a 4 a 4 is nothing but a again multiplied to 3a minus 2i Which is nothing but 3a square minus 2a And a is once again a square is once again 2a minus i So that will give you 4a minus 3i So further it emphasizes the fact that if you continue doing that a to the power n and being some positive integer will become na na Minus n minus 1i Please start this question. This is a very very famous question. It comes in many combative exams If p is an orthogonal matrix And q is pa p transpose And x matrix is p transpose q to the power thousand p Then x inverse is a is involuntary over here. Please note a is involuntary Multiple options can be correct Okay Last 20 seconds, then I'll discuss it. Please wrap this up Okay, let's discuss this See it q is pa p transpose And p is an orthogonal matrix right Now if you find the matrix x basically, what are you writing? You're writing P transpose q to the power thousand q to the power thousand means you're writing this stuff thousand times correct If p is orthogonal you realize p into p transpose will be i And again p into p transpose will be i again If you keep on doing it, basically you'll end up seeing that It'll form a chain like this Okay, so even this will become an i Even this p with a p transpose before will become an i like this. So it'll be i a i a i a like that Okay, so it'll become a to the power thousand Right now if a is an involuntary matrix Then can I say this will be as good as i and it'll be as good as As good as itself Involuntary matrix is basically a square is equal to i correct I do it correct Yes, sir any any power raised on an involuntary matrix Okay, so a q will also be a A4 will also be i and so on so a will be i only So I think b option is correct And c option is correct. Is that fine? Next question Metrices is not as easy as you thought it to be Yeah, yes Now just remind me I will just clarify that Item potent questioned out a plus b will be item potent when a a b is equal to b a is equal to null I I've given enough time for you to respond I'll clarify that before I close the session. Oh, sorry. I forgot to put the poll on Let's have the poll Why not we can have variables No, we cannot differentiate a matrix, but we can differentiate a determinant We can differentiate the terminate we can integrate the terminate Oh Oh, yeah, and I thought yes, and I thought please you're most welcome to clarify it So and I thought we'll clarify that question if you left. Sorry Venkat. You were saying something Oh, yes, sir, like do we integrate determinants and all my expansion and then There is a property which we'll talk about Of course expansion you can always apply Okay, uh, we'll close the poll in another 40 seconds of those who have not responded please do that I have a slight doubt in the answer. Are you getting 2010 over here? Anybody? Yeah, yes Anyways, we'll see we'll see uh, if you feel that none of the options are correct Then do let me know on the chat box. Okay. Okay. Let's let's uh end the poll and discuss it. I think Okay, we'll see we'll see whether this whole result is correct or not because I have a slight doubt in the options. We'll check it out See if a and b are square matrices satisfying ab is equal to ba Sorry, ab is equal to a and ba is equal to b. What does it mean? That a and b are important, okay the moment you see this The the property should come in your mind that a and b are idempotent Okay If n be a idempotent, we know that in idempotent matrix if you raise any power it will be the same matrix right ultimately it will be This only correct So these powers are just carrying you off. It is just a a plus b actually so it's a plus b to the power 2011 Correct does everybody agree so far with me That a to the power 2010 and b to the power 2010 are nothing but a plus b only Okay, now what I'm going to do here is I'll just do a simple pattern check I'll do a plus b square Okay, and this gives you a square plus b square Plus ab plus ba Which is back to Which is back to a plus b and ab is a and ba is b. So again a plus b. So you'll get two times a plus b Okay, if you continue with this that means if you do a plus b to the power, let's say, um three You'll end up getting two a plus b square Which is back to two a plus b so two into two a plus b So what I'm seeing here is that This a plus b doesn't change but this power is one less than this index correct So since you're going till 2011 I doubt whether this will go to 2011 it'll go to 2010 So option d is correct, but with a minor correction over here. So please make a change Uh, sorry, I didn't uh For see that So could you do this using binomial theorem? But for binomial ab and ba should be equal For binomial expansion to work ab should be equal to ba So who told that uh a and ba are equal There's no there's no such Like how how you've uh Used like you've started with two and three. Could you just expand this using binomial theorem and collect all the terms No, we can't do binomial theorem. No, we can't write this as a square plus b square plus two ab This cannot be written like this That will only work when ab and ba are same. That's what I'm trying to say Yes Okay, yeah now, uh, anita Can you just explain that question? Can we go back? I don't know which slide number was that also We can do it separately. I mean it doesn't take much of a time So the property was a and b were idempotent and the question was a plus b will be idempotent if and only if ab is equal to ba is equal to null Yeah, anita, can you just unmute yourself and say So yes, sir start with this discussion that we reached here after this what's Yeah So if you multiply b On the right side post facto Okay post facto b here. Okay So if you know that ab square plus ba b zero is null Yeah, yeah, and you can write b square as b because very good That means you can write ab as a minus ba b very good Yeah, carry on. You're on the right track And then you multiply pre facto ab plus ba Again multiply pre facto With a multiply ab plus ba again. Okay. So can I do one thing? Oh, sorry. What what what did you say? I multiply ab plus ba Pre facto null by b Uh, you can do one thing you can multiply pre facto with a b. No Yes, so that's what I said us. Yeah. So now this will be b again Yeah, and you get very good Correct. So ba is also minus ba b. Correct. So basically she proved that these two are equal Because both are equal to the same expression, right? correct now Putting it back over here if ab and ba are equal it means each has to be equal to The null matrix very good and either Awesome. Is this fine? all right, so Let me tell you we are just done with half the topic of matrices other half still will take four hours I'm sorry. I had a misjudgment about the length of the concept. So it will take a small time to do it We have to still do symmetric skew symmetric Hermitians skew Hermitian Uh, elementary transformations Then we have to do a martin's rule And we have to do yeah Mostly that's it and then we can start with A determinants concept Anyways Thank you so much. I'll stop here Hope you have now a different perspective about matrices So it's not a chapter to be done on your own You definitely need some kind of guidance here Okay, thank you. Have a good night. Thank you, sir. Thank you, sir. Thank you. Thank you, sir. Thank you, sir Thank you, sir. Thank you