 Hello, good afternoon. Welcome all to the YouTube live session on matrices Those who have joined in the session, I would request you to type in your names in the chat box so that I know who all are attending the session Hello, okay. So guys, please type in your names in the chat box so that I know who all are attending the session and The biggest prerequisite for this session is have you watched the Videos sent to you on WhatsApp Recorded for matrices So a question is have you all seen the videos been? Yes, if you have seen it and know if you haven't Bharat Shankar has not seen I haven't seen all sir. Okay all right Okay, I think you have done Matrices to a certain extent in your school, right and This is not a completely new session for you. You have some prior experience with matrices So since this is a J advance session, I would not spend time talking about what is Matrix addition. What are the types of matrices? Of course, I'll spend some time on talking about special matrices But at least not about addition Or multiplication or scalar multiplication of matrices. I'm assuming that you all are aware of that Okay, so I'm going to begin this session with special matrices. I'm assuming you are aware of null matrix you're aware of Collar matrix row matrix Scala matrix diagonal matrix, etc. So I'm going to just start with the special matrices and In that I'm going to start with symmetric and skew symmetric Symmetric and skew symmetric How many of you have absolutely no idea what's a symmetric matrix and what's a skew symmetric matrix? Just type me if you don't have any idea about symmetric and skew symmetric Okay, so I'll quickly recap this all if you know transpose of a matrix, right? At least I don't need to tell you what's a transpose of a matrix correct So any matrix a if it is a square matrix Let a be a square matrix which satisfies a Transpose is equal to a Then this matrix is called a symmetric matrix Okay, whereas If a let me not write and in between If a satisfies this a transpose is negative of a then we call that as skew symmetric matrix correct Let me give you a simple example of a symmetric matrix Do you remember the delta that we used to have in our a pair of state lines? Which we had as a bc Hg Hg f f This is a typical example of if you make a matrix out of it Okay, then this basically becomes a example of a symmetric matrix Okay, on the other hand if you have something like this, let's say a sorry Zero zero zero H g f and this is a minus H minus g minus f Then this is an example of a skew symmetric matrix Now why the elements in the leading diagonal position? This is called the leading diagonal position of the matrix or the principal diagonal You'd see that in case of skew symmetric matrices all the elements in the leading diagonal position would be zero Why is that so? Why is that so any idea right? See basically what you're trying to say if you go by this definition You're trying to say that a i j a i j is is Negative of a j i isn't it Where a i j is the element in the ith row and jth column position of the matrix So if you make i is equal to j that means if you're referring to elements if you're referring to elements in The leading diagonal position in the leading diagonal position you realize that it becomes a i i is equal to negative a i i That means to a i i is equal to zero that means a i i will be equal to zero and Therefore, it's a very important property You start irrespective of that you can start So you can see that the elements positioned in the leading diagonal position should all be zero for a skew symmetric Not only that you would see that in these two matrices in In case of a symmetric matrices the elements Which are mirror image about the leading diagonal are all same see h and h are same g and g are same f and f are same But in case of a skew symmetric matrix the element in the leading diagonal positions would be of opposite sign So as you can see h-h g-g f-f Yeah, yeah, yeah, we'll cover all of those things Don't worry Bharat Okay Now what are the properties of symmetric and skew symmetric matrix? These are very important because a lot of questions have been asked on properties of symmetric and skew symmetric matrices Symmetric and skew symmetric matrices first property is if a is any square matrix if a is any square matrix then a Plus a transpose this will always be a symmetric matrix This will always be a symmetric matrix Okay, hi Shia good afternoon everyone why are people coming late? I Think your school got over by 12 right 12 30. Okay. Why do the symmetric matrix very simple? You can prove this if you're claiming that this is a symmetric matrix. Let's say I call this as C Right, you can realize that you can easily prove that C is equal to C transpose Okay, for that you need to be well aware of the properties of transpose of a matrix, which I'm sure you would have seen in the videos Okay one few important properties of transpose is that I Just quickly recap for people who have not done this before properties of transpose first property is Transpose of a transpose will give you the same matrix back second property is This would be k times a transpose a plus B transpose would be a transpose plus B transpose a B transpose is B transpose a transpose this law is called This law is called the law of reversal next is a Power n transpose where n is some positive integer is as good as a transpose to the power n and a inverse transpose and a transpose inverse both are the same, okay Yeah, so using this property now. I'm going to prove that C is equal to C transpose So let's start with our C transpose C transpose will be a plus a transpose transpose Now I will use this property to write it as a transpose plus a transpose transpose Which will become a transpose and by this property. I can write this as a And you know matrix my addition is commutative you're back to see again Which implies that this matrix will always be a symmetric matrix Okay, similarly a minus a transpose will always be skew symmetric a Minus a transpose will always be skew symmetric, okay? Second property is sorry. Let me continue with this only a Into a transpose will be what? Will this be symmetric or skew symmetric? What can you comment about a into a transpose? Will it be symmetric or skew symmetric? That also will be symmetric, right? It's very simple to prove if you do a transpose transpose It will become a transpose transpose by the law of reversal Which is back to a into a transpose Okay, so this will always be symmetric. Is that fine? Oh, let me just clear this off next property property number two if A is symmetric remember following things Minus a will also be symmetric Minus a will also be symmetric. In fact any scalar multiplication with a will be symmetric Okay, a transpose will be symmetric a to the power n where n is some positive integer that will also be symmetric a Inverse will also be symmetric. Is that fine now? In a similar way If a is skew symmetric, can I say minus a will also be skew symmetric Minus a will also be skew symmetric. In fact k a will also be skew symmetric, okay? a transpose will also be skew symmetric Okay, now a to the power of an odd power will be skew symmetric But a to the power of an even number will be symmetric Can you prove this? Can you prove this? Can you prove that if a is a skew symmetric matrix a raise to the power To n plus one of course n is a positive integer here will be skew symmetric Whereas a raise to the power and even number will be symmetric Just do the proof for it and send me the picture of it on WhatsApp. Okay, it's very simple. Let a transpose be negative a Okay, since it is a skew symmetric matrix Correct. Now I want to know what is the nature of a to the power 2n plus 1 So what I'll do is a to the power 2n plus 1 I'll transpose it first So that is as good as a transpose to the power of 2n plus 1 correct Which is as good as saying minus a to the power of 2n plus 1 correct now, this is as good as saying that it's minus 1 to the power of Minus 1 into a to the power of 2n plus 1. Yes or no Remember a matrix minus a multiplied to itself odd number of times would leave you a minus sign outside Okay, so this matrix is Going to be skew symmetric in nature because you started with the matrix C Let's say I call this as a matrix C Right, so this is C transpose And you ended up with a minus C. So C transpose becomes minus C. So this matrix is symmetric And at the same time, it's needless to do that when you raise it to the power of even integer You're multiplying minus a even number of times And that would leave you with only a C and therefore, this is a symmetric matrix Now property number four, which is important for your school exams. I just now mentioned two things That if a symmetric Sorry, if a is a square matrix a plus a transpose Is going to be a symmetric matrix correct and a minus a transpose is going to be a skew symmetric matrix Okay Now can I say Half into a plus a transpose this will also be symmetric This will also be symmetric because I just not told you k into any symmetric matrix will also be symmetric Right, similarly, I can say half into a minus a transpose will be skew symmetric correct What happens when you add them? Let me call this as p for the time being let me call this as q for the time being What happens when you add p plus q? When you add p plus q You get something like this Which is nothing but which is nothing but your matrix a itself Okay, which implies that Any square matrix Any square matrix can be expressed as a Sum of a symmetric plus a skew symmetric matrix Where symmetric matrix here is p and skew symmetric matrix is q yes Yes, right girl. Is that a lot of questions way to come on this in your school exams Whenever a question a matrix is given to you all you need to do is do a plus a transpose half That would be your p part. This would be your p part Okay, and if you do a minus a transpose half that would be your q part And this is what you need to show in your school exams okay Next property property number five if a is symmetric and B is also symmetric. Please note down that a plus b or a minus b both will be symmetric a b will be symmetric if and only if The a b multiplication is commutative. Please remember this This is very important very obvious. We can prove this Let's say C is your a b correct, and you know a and b both are symmetric If you do c transpose it is a b transpose which is nothing but b transpose a transpose according to law of reversal Okay, and b transpose a transpose is b a and if you want this to be a b back Which is actually c then these two must be Following this criteria Moreover a b plus b a will always be symmetric a b plus b a will always be symmetric Right again, I've already proved this in my video lecture on matrices So you can access that let's say I call this as c. So if you do c transpose It's a b transpose plus b a transpose Which is nothing but b transpose a transpose Again, this is a transpose b transpose Which is nothing but b a plus a b back because a and b are symmetric matrices and Since matrix addition is commutative you can write it back like this which is equal to c So c transpose and c both mean the same thing. So this is always a symmetric matrix on the other hand if a is q symmetric and B is also skew symmetric a plus b a minus b Both will be skew symmetric a b minus b a Will also be skew symmetric Okay, by the way Let's add one more thing over here a b minus b a in this case will also be skew symmetric Despite the matrices being both symmetric a b minus b a in both the scenarios In both the scenarios will be skew symmetric Okay, and a b plus b a in both the scenarios will be symmetric. It's very easy to prove Let's say a And we both are skew symmetric This is skew symmetric and B is also skew symmetric. Okay, if you do a b plus b a transpose Okay, let's say I call this as c transpose you get a b transpose plus b a transpose Which is b transpose a transpose plus a transpose b transpose which is negative b negative a Again negative a negative b which is again b a plus a b Which is again a b plus b a because it is commutative and you get the same matrix back So this will always be symmetric right So this is a very important property despite a and b both being symmetric or skew symmetric a b plus b a will always be symmetric and At the same time a b minus b a will always be skew symmetric. Yeah, let's check. Let's check that Well, let's check that so if a b minus b a is Transposed what do we get we get a b transpose minus b a transpose? Okay, now I'm assuming that let's say the matrices a and b both are symmetric Okay, so this will become b transpose a transpose This will become a transpose b transpose. Yes or no, isn't this b a minus a b? Isn't this negative of a b minus b a? Isn't it minus c then let's say I call this as c transpose So c transpose is giving you c back which implies c is skew symmetric Got it away Okay next property number seven if a is symmetric if a is symmetric Please note b a b transpose if at all this matrix multiplication is conformable This will also be symmetric where b is If you can call it a square matrix or you can call it such a matrix Which is conformable to be multiplied with this Okay, so you can say it's to be a square matrix Okay, need not be a square matrix. It can be any matrix which is conformable to be multiplied or any matrix conformable to be multiplied or any matrix conformable To be multiplied all of you know conformability test for multiplication, right? The number of rows the number of columns of the brief Factor should be same as the number of rows of the post factor. Okay And same will be true. Even if it is b transpose a b. This will also be symmetric Provided it is conformable to be multiplied okay similarly if Let me call this as property eight If a is skew symmetric Okay, then b a b transpose will also be skew symmetric and b transpose a b This will also be skew symmetric. Is that fine? so these are the properties of Symmetric and skew symmetric matrices that we need to know now time for some questions Let's have some questions my first question to you is find the find the maximum number of different elements Find the maximum number of different elements required to form a symmetric matrix of Order 12 Yes, yes law of reversal law of reversal is true for more than two matrices also for example Let's say if you do a b c d transpose It's as good as d transpose c transpose b transpose a transpose Okay Law of reversal works for transpose. It works for inverses also Okay, it's c inverse b inverse a inverse It works for adjoint also. It's the adjoint of c Into adjoint of b into adjoint of a it works for Conjugate transpose also. I'll talk about all these things in today's class. Don't worry Okay We'll discuss these we'll discuss these don't worry. We'll discuss Okay, so I already have the answer so people are saying 66 78 Okay, it's very simple that if you have a 12 by 12 matrix all the elements in the diagonal position You need them so they are 12 those 12 of those correct all together. They are total number of elements total number of elements would be 144 Okay, 12 into 12 which is 144 But the thing is that since you are making a symmetric matrices the element in the a ij position is same as the element in the ji position, right? So basically whatever are the remaining elements you can divide it by 2 Okay, and add a 12 to the answer so this would be your total number of different elements Different elements required so that is going to be 132 by 266 plus 12 that is 78 the 78 is your answer next question x and y 3 by 3 square matrices 3 by 3 non-zero skew symmetric non-zero skew symmetric matrices and Z be a 3 by 3 non-zero Symmetric matrix then which of the following matrices? then Which of the following skew symmetric option a y cube z to the power 4 minus z 4 y to the power 3 Option b x to the power 44 y to the power 44 option c x to the power 4 z cube Minus z to the power 3 x 4 option d x to the power 23 plus y to the power 23 please type in your response in the chat box. Yeah, it's a multi correct. It's a multi correct. Yeah DC and a C and e okay. I'm getting various types of answers. So let us discuss this Let us take the first one so if you do y cube z to the power 4 Minus z 4 y to the power 3 transpose Okay What do we get we get y to the power 3 z to the power 4? Transpose minus z to the power 4 y to the power 3 transpose Correct you can write this as z to the power 4 transpose We can write it as z transpose to the power 4 y transpose to the power 3 minus Y transpose to the power 3 z transpose to the power 4 Okay So remember one thing that is a symmetric Matrix, so this is as good as z to the power 4, but why is this Q symmetric matrix? So it becomes Y cube Okay, here also I get a minus y cube And this becomes z to the power 4 Okay, it becomes minus z to the power 4 y q plus y cube z to the power 4 Which is same as the original matrix correct? So those of you who said it is a skew symmetric you are wrong because this is a symmetric matrix Okay, so a cannot be the answer. Is that fine? God of away Next let's check the other one Now x and y this has to be symmetric because the power is or even number on both of them So this cannot be the answer. Okay Let's talk about this So x to the power 4 z cube minus z to the power 3 x 4 Transpose What will it become? Again x to the power 4 z cube transpose Minus z to the power 3 x to the power 4 transpose Again law of reversal will work law of reversal will work correct Okay, now since x is a skew symmetric matrix. It will remain X only because of the power 4 so it is as good as saying z cube x to the power 4 Whereas here it will become again same x to the power 4 z cube Isn't this negative of the original matrix? It's negative of the original matrix So this is a skew symmetric matrix option C is the right option Okay, and D has to be because it has got odd powers on it So the right options here is C and D are the correct options in this guys Come on. You can't get the first problem wrong on this next question a3 a5 and So on till a2n minus 1 Our n skew symmetric matrices are n skew symmetric matrices of Same order of same order given by summation of 2r minus 1 a2r Minus 1 to the power of 2r minus 1 will be option a symmetric q symmetric Neither symmetric Norse q symmetric and option D Data not adequate. Yeah option B everybody saying that anybody who differs from B. Yes So B is the right answer. Let's discuss it quickly. So basically you're saying B is nothing but a1 plus 3 aq a3q that Plus 5 a5 to the power 5 and so on till You reach This so a1 remains a1 In fact, if you transpose it You are transposing the entire stuff and so on till so this becomes minus a1 Because it's given that a1 is a skew symmetric matrix. This becomes minus 3 a3 and So on till the last becomes minus 2n minus 1 a 2n minus 1 to the power of 2n minus 1 Okay, which clearly implies this is minus of B Therefore B is going to be a skew symmetric matrix next question is if s is a skew symmetric matrix or order 7 by 7 Okay, I can say just order 7 Determinant of s is equal to is the data sufficient for you to answer this question. Exactly. It will always be 0 Right Can somebody prove this a skew symmetric matrix if a is a skew symmetric matrix Let me write this if if a is a skew symmetric matrix of odd order then Determinant of a will always be 0 prove this. Yes, it's a it's so in case of all skew symmetric matrices Can you prove this and send me the a? Pick of that proof to my personal whatsapp. Can you send me the pic? Hmm. Okay. See we all know that Determinant of a is same as determinant of its transpose Okay, if you're not aware of this, please remember this. Okay, it's the property of determinant correct That means we're trying to say determinant of a is Determinant of negative a now please recall that if a is of if a is a square matrix of Order then determinant of k a will be k to the power m determinant a now if you apply the same over here assuming that your K here is minus one. So this is your k Right, and you know a is of odd order. Let's say the order is 2n plus 1 So I can write it as minus 1 to the power 2n plus 1 Determinant a which is actually negative of determinant of a so determinant a is negative of determinant a which means to Determinant a is 0 that means determinant a is equal to 0 always Okay But what can you say about determinant of skew symmetric matrices of Even order what about determinant of a if a is of even order can you comment Can I say they will always be a Positive number Can I make such comment? Can I say it'll be greater than equal to zero always? Yeah, greater than equal to zero. That's what I'm saying. See I take a simple example Let's say I have a skew symmetric matrix of order 2, right? What is the determinant of this 0 minus of Minus a square right, which is a square is always greater than equal to 0 Yes or no Okay. Now, this is a homework for you Try to see whether Thus determinant of a skew symmetric matrix of even order is it always greater than equal to zero or can you find some exceptions? guys moving on to the next type of special matrix which are I have not covered in those videos is conjugate of a matrix So what is conjugate of a matrix conjugate of a matrix is basically nothing, but If you take the conjugate of all the elements right for example, let's say for example, let's say I have a Matrix a like this Let's say 3 minus 4i 7 2 minus 3i and Minus 1 minus 2i, okay Conjugate means you are trying to find the conjugate of every element in this. So basically you are trying to do this This is a conjugate of the Matrix a Okay Now the conjugate itself is not important But there is a concept related to it, which I'll be coming shortly to that is that is going to be very important for you Meanwhile simple properties that you should know of Of conjugate simple property related to conjugate of a matrix First one conjugate of a conjugate returns the same matrix back, okay Secondly a plus b if they are conformable to be added a plus b conjugate will be a conjugate plus b conjugate, okay? secondly a B conjugate if it is conformable to be multiplied will be a conjugate b conjugate This does not follow the law of reversal This does not follow the law of reversal Okay, this is the only case where law of reversal will not hold true While in many other cases like transpose inverse as I told you adjoined they all follow the law of reversal Okay, and if you multiply alpha a Alpha b alpha can be a complex number also Okay, and if you take its conjugate it will be alpha conjugate into a conjugate Okay What is important for us is? transpose conjugate of a matrix transpose conjugate of a matrix if you conjugate and transpose it we call it as a special term This or many books will call it as a star Okay, for example For example, if I give you a matrix like this a 2 plus I 1 3 1 2 3 plus 2 I 3 3 plus 2 I Let's say 6 minus 7 I okay Let's say I take minus 3 I over here and 2 I over If I ask you what is conjugate transpose or what is a theta or a star What you'll do is you start conjugating and transposing it so 2 minus I 1 3 I 1 minus 2 I 3 minus 2 I 3 3 minus 2 I and 6 plus 7 I this would be your conjugate transpose Okay, let us look into some properties of this properties of Conjugate transpose or transpose conjugate of a matrix Yeah, you're doing both the things here. You're transposing it also and you are taking the conjugate of every element first property is if You transpose conjugate a matrix twice you'll get the same matrix back second is if you add two matrix and do the transpose conjugate it is as good as this Secondly if you multiply a Matrix with a Number k k can be complex also k can be a complex number also Okay, you will get k conjugate Times a transpose conjugate Okay, and Finally the law of reversal which I discussed with you earlier also a into b conjugate transpose will be This Now what is important for us here are the two types of matrices associated with it which we call as Hermitian and skew Hermitian matrices. What are Hermitian and skew Hermitian matrices a square matrix a a Square matrix a if it satisfies the criteria that a Is equal to a transpose conjugate then it will be a Hermitian matrix It would be a Hermitian matrix If it satisfies this criteria Then it would be called as a skew Hermitian matrix It would be called as a skew Hermitian matrix Now couple of things that I would like to ask you couple of things that I would like to ask you if a has all real entries if a has all real entries right Can I say Hermitian Would be same as Symmetric matrix or not? Yes or no because ultimately what is a star doing a star is basically conjugating and transposing right? But if a already has real entries Can I say a star would be as good as a transpose? Because conjugating will have no influence Correct. So in that case when you say a is equal to a star for a Hermitian or a theta for Hermitian You are indirectly saying a is equal to a transpose Yes or no Yes or no, okay Second thing that you would observe that if a is a skew's Hermitian matrix if a is a skew Hermitian matrix all its entries All its entries In the leading diagonal position Will either be zero purely imaginary. So one is this property another is this property Okay Can you prove this? So what we're trying to say is that if a i j is Negative a j i conjugate Right and if you put i is equal to j it means you are saying a i i is negative a i i conjugate which implies a i i Plus a i i conjugate is equal to zero this can only happen if Only possible if a i i is either zero or a i i is purely imaginary Now something very interesting here also since we have talked about real entries Can I say all skew Hermitian matrices will be same as skew symmetric? I think I forgot to mention that yeah, so Gaurav has talked about trace Yeah, what is trace? All of you know trace What is trace of a matrix? Trace of a matrix is nothing, but It's it's for a square matrix It's nothing, but the sum of all the elements in the leading diagonal position I'll come back to a question on trace. Okay. Thanks for reminding me about trace We have a lot of interesting questions coming up on trace also Okay, since we have talked about trace. It's very important to see the properties So let's let's talk about properties of trace first property is a trace of any Square matrix multiplied to lambda a scalar quantity will be as good as Lambda times trace of a Okay Secondly trace of a plus b is as good as trace of a plus trace of b Okay And trace of a b here also you see the law of reversal Will hold true trace of a b and trace of b are the same Trace of a into a transpose will always be positive There was a very interesting question based on this So I'll just take a question before I go back to the Hermitian and skew Hermitian I have not yet finished off, but there's an interesting question on this which came a couple of years back Let's say we have a matrix a Which is two one four one There's a matrix B which is let's say three four Two three and there's a matrix C which is three minus two minus four three Find the sum of trace of a trace of a b c by two trace of a b c square by four Trace of a b c cube by eight and So on all the way till infinity He has already done these questions Why why are you doing this to us before the question is completed you're giving the answer Barat is a spoiler today. Okay. Others. Please try ignore his answer. Oh My god. Yeah this okay. So you all realize that only BC is getting squared in cube, so let us find BC first BC will you what you all know matrix multiplication? Row of this multiplied to column of this so three into three minus nine minus eight this will be minus twelve plus twelve this will be 6 minus 6 and this will be minus 8 plus 9 so that is going to be 1 0 0 1 correct right so if you see this term trace of A the other would be just simply trace of A by 2 then so this term will be just trace of A by 2 this term will be trace of A by 4 and so on okay till infinity now I have already discussed with you the property over here that if you multiply any matrix with a scalar quantity and find its trace it is as good as multiplying the scalar quantity with the trace of A so let us apply that so it will be as we were saying trace of A plus half of trace of A plus one-fourth of trace of A and so on so it's 1 plus half plus one-fourth all the way till infinity times the trace of A so trace of A is here is 3 right this is the trace of A trace of A is 2 plus 1 which is 3 so it's going to be 3 times 1 by 1 minus half that's going to be 6 okay fine so I have another question based on Hermitian and skew Hermitian matrices question is show that for every square matrix A A can be uniquely expressed as p plus iq okay where nq are Hermitian matrices is the question clear so you have to show that any square matrix can be uniquely this expression is unique this is a unique expression you cannot have two different sets of Hermitian matrices satisfying the same so prove that if a is a square matrix if a is a square matrix a can be uniquely expressed as p plus iq where p and q are Hermitian matrices yeah elements in a can have real and imaginary components any idea just a question for you let's say a is a square matrix with the entries of a being complex also can I say a plus a transpose conjugate this will always be Hermitian can you prove this quickly we can directly do this transpose conjugate and by the property I can write it like this it is as good as saying this and we are back to the same matrix right so if you call this as C then we have proved that C transpose conjugate is C correct okay in a similar way I can say if a is Hermitian let's say not a let me call it as some different names so that you don't get confused if B is Hermitian k into B will also be Hermitian again how do I prove this same if let's say I call this as D then D transpose conjugate you can say it will be this into this correct right and if k happens to be real number right it is same as k B back okay which implies that from all these two findings over here that half a plus a this will always be Hermitian okay is that fine no doubt about it now what about a minus a transpose conjugate let's say I call this as C so if you do transpose conjugate of this you get this okay and let's say you can write it like this which implies that you're getting minus C back correct therefore can I say this will always be skew Hermitian yes or no now if you multiply a skew Hermitian let's say if B is skew Hermitian okay can I say k B will also be skew Hermitian provided k is real number right again we can prove this very easily if you take this it becomes conjugate this okay and this will be k back and this will be negative B okay so this is negative of the original answer so this is going to be skew Hermitian for sure is that fine now take this as a hint and try to show that a can be expressed as P plus IQ and find the P and Q for me if you're not able to get let me know I'll be helping you even further I'll give you more hints if let's say D is Hermitian then what can you comment about then what can you comment about a complex number let's say a purely complex number let's say I let's say lambda I into D let's say I into D not lambda I let me just remove this lamp what can you comment about I into D will this be Hermitian or will this be skew Hermitian let me change this question here let's say this is skew Hermitian what can you comment about ID it's very simple just do a small check ID conjugate trans transpose conjugate will be I bar D this and if D is skew Hermitian this will become minus I and this will become minus D correct isn't it back to ID so can I say this will be Hermitian correct my light of all these things can I say that a can be written as half a plus a transpose conjugate plus I times 1 by 2 I a minus a transpose conjugate yes or no as you can see this will be Hermitian correct and this will also be Hermitian so let me call it P for the time being let me call this P for the time being and this has skew for the time being so isn't like a expressed as P plus I Q yes or no so please remember this here P and Q both will be Hermitian remember if I just write half a minus a transpose conjugate this is Hermitian correct sorry this is skew Hermitian but the moment I write an I over here it becomes a Hermitian are you getting it so this is what has happened over here by the way in the previous example I multiplied I on the top if you multiply I bottom also doesn't make a difference we can quickly check we can quickly check let's say if E is Q Hermitian let's say if E is Q Hermitian okay that is E transpose conjugate is minus E and you want to know the nature of E by I okay so let's do this so it is as good as 1 by I this into this right which is 1 by I bar and this is negative E I bar is negative I which is back to E I okay so E I has now become a Hermitian matrix okay so that is what over happened over here so this was Q Hermitian the moment you divided it with I it became a Hermitian is that fine let's take other special matrices then we can start solving problems orthogonal matrix the matrix a which is a square matrix it is orthogonal if it satisfies a transpose is equal to I that is you're trying to say that a transpose is same as the inverse of that particular matrix okay next is idempotent it has to be a square matrix first of all it is idempotent if it satisfies a square is equal to a is that fine correct yes what will be a cube then can I say a cube will be a square into a which is again a into a which is again a square which is again a back yes or no correct yes or no what is a to the power 4 a to the power 4 is a square square which is a square itself which is a back again yes or no so please note that this is the property satisfied by idempotent matrix I'll do that we'll come to that so Keith will come to those transformations next is pollute matrix again it is applied to a square matrix it will be involuntary if it satisfies a square is equal to I that is you're trying to say the matrix is its own inverse this connection timeout what happened has internet been disturbed check just check the internet what is it can you see the screen also yeah yeah now it is back can you see the screen okay great okay fine it is applied to a square matrix a square matrix if it satisfies this property that till a to the power let's say m minus 1 the matrix is not a null matrix but a to the power m is becoming a null for example let's say I start with a okay this is not null I did a into a which is nothing but a square this also is not null right now I did a cube this also is not null now let's say a for has started becoming null then this is called a nil potent matrix of order for this will be called as a nil potent matrix nil potent matrix of order for yeah yeah you can see the screen in sometime just hold on so this you would call as a nil potent matrix of order m is that fine let's take few questions on this so we have been talking so much about the theory let's talk about some questions by the way I'll give you some examples of these matrices for example orthogonal let's take a question on orthogonal if s is a skew symmetric matrix i minus s into i minus s inverse is orthogonal if s is a skew symmetric matrix a minus s sorry i plus s i plus s I am sorry it's not i i plus s by i minus s inverse is orthogonal yeah the product the product of i plus s and i minus s inverse is orthogonal yes Ashutosh I'll be doing the properties of adjoint inverse everything don't worry about it I've still not done with the types of matrices then I'll talk about a bit of polynomial operation on matrices and then I'll come to those adjoint and inverse questions proved okay let's say I call this matrix as a so a is i plus s and i minus s inverse so what is a transpose a transpose will be i plus s and i minus s inverse transpose right which follows the law of reversal which means i inverse transpose into i plus s transpose correct now please note inverse and transpose positions can be stopped so it becomes i minus s transpose inverse and this becomes i transpose plus s transpose okay here also I can say i transpose minus s transpose inverse and i transpose plus s transpose now note that I even if you transpose it remains I s transpose will be negative s because s is a skew symmetric so it implies s transpose is negative s correct so this becomes this and this becomes i minus s correct now if you do a into a transpose what do we get okay a was what a was i plus s okay or you can do a transpose into a also a transpose into a a transpose into a means i plus s i minus s let me do the other operation I think a transpose was the correct operation a a transpose okay so this was your okay a a transpose however written everything correctly just check i plus s inverse i minus s a into a transpose so a into a transpose okay this is correct yeah why am I getting the opposite science here a transpose into a would be what a transpose in this into i minus s i plus s and this inverse correct now i plus s and i minus s is it commutative can I show that this is as good as saying i minus s into i plus s okay let's do that first so it's i minus s plus s minus s square right this is your LHS what's your RHS similarly i plus s minus s minus s square so they are same so what I'm going to do is I'm going to flip the positions of these two okay if I flip the position of these two it becomes i plus s i minus s and i minus s inverse correct now in this if you see these two will become an i these two will become an i i into i will be i yes or no so here you have to show briefly that the matrix multiplication here i plus s and i minus s is going to be commutative so because of this commutative nature I could swap the positions of the middle two and hence I got the answer as I is that fine okay next question in which of the following type of matrices inverse does not exist always for which of the following type inverse does not exist always options idempotent orthogonal involutary none of these idempotent I told you you know idempotent is a square is a this is idempotent when a square is i this is involutary none of these okay guys all of you first know this property that if a and b are square matrices determinant of ab is as good as determinant of a into determinant of b correct now if you say idempotent that means you're saying if a square is equal to a or a into a is equal to a let me take the determinant on both the sides so this I can write it as determinant a into determinant a is equal to determinant a correct which means determinant a can either be one or determinant a can be zero correct now the moment the determinant of a matrix becomes zero if you would have seen the videos it becomes a singular matrix singular matrix are not invertible okay not invertible means there would be no inverse existing for it so option number a is definitely correct orthogonal what is orthogonal orthogonal means a into a transpose is i so if you take a determinant that means determinant a into determinant a transpose is going to be one and this and this are same things because a transpose determinant and a determinant are same that means this is equal to one which means determinant a is plus minus one so this can never have determinant a as zero so this cannot be my answer okay by the same logic involutary also cannot be my answer so option number a is correct does not always does not exist always there may be a case of zero also getting it Bharat next question if p is orthogonal matrix if p is orthogonal and q is equal to p a p transpose and matrix x is p transpose q to the power of thousand p here a is a involutary matrix x inverse is option a a option b i option c a to the power thousand option d none of these so keerth has given the response what about others Santosh also says b or c okay not sure yeah it can be multi-correct also okay let's check this so now if you're doing this operation let's say p transpose let me write in white p transpose q q is again p a p transpose and you're writing it thousand number of times p a p transpose p a p transpose till p a p transpose and it ends with a p okay now we all know that p is an orthogonal matrix that means p p transpose will be i so this will be i a again this will be i a again this will be i a and so on till you reach this so ultimately we'll be writing a thousand number of times but at the same time i is involutary right remember i told you a square is i a cube will be a a four will again be i correct so if you have a to the part two n your answer will always be i so this will also be i so both the options are possible for x okay so x inverse will be i yes or no and i itself is a to the power thousand so both option b and c will be correct is that fine next question if z is an item potent if z is an item potent matrix i plus z to the power of n okay n is a positive integer is option a i plus two to the power n z option b i plus two to the power n minus one z option c i minus two to the power n minus one z and option d nota suki i'm sure you would have scammed it haven't you suki this scammed it right okay let's let's look into this it's not a very difficult question see actually when you're talking about this no you can apply binomial theorem on this okay so you can write it as nc zero i to the power n correct nc one i to the power n minus one z nc two i to the power n minus two z square and so on till ncn i to the power zero z to the power okay yeah but remember binomial works only when the matrices involved are commutative in nature for example let's say ab is equal to ba ab is equal to ba then only i can write a plus b square as a square plus two ab plus b square that means binomial theorem is working here okay binomial theorem works fine on this right so z and i they follow commutative nature so i z and z i are same thing so it doesn't matter to us okay that's why we can do all these things there's something which is very important but if they're not commutative binomial cannot be applied so if ab is not equal to ba please remember you have to write a plus b whole square as a square plus ab plus ba plus b square like that is that fine so here binomial theorem will not work oh my god we are missing this term uh sukih symmetrical okay so now when you do this it is as good as saying i right i to the power n is i only right so this is going to be uh this is also going to be z okay nz in fact or nc one z then you have nc two z square but since it is an idempotent matrix what will happen z square is going to be z even z cube is going to be z right and so on we discussed about that so i can say that all these terms will start becoming z nc three z all the way and this will become ncn z okay so what you can do is you can take z common nc one nc two all the way till ncn and that is going to be two to the power n minus one we have already studied that so option b is correct option b is correct is that fine next question let f of x be one minus x sorry one plus x by one minus x okay and a is a matrix which satisfies a cube is equal to a null matrix oh okay then f of a is option a i plus a plus a square i plus two a plus two a square i minus a minus a square and option d none of these what however it now it's not a division that you have to do you have to write it like this a plus a into i minus a inverse this will be your f of a okay this is something which i'm going to talk next okay what about others guys okay let's let's solve this as i told you when you have two matrices involved where matrix multiplication is commutative you can apply binomial theorem on it right just give me one minute this is zoom from where we left off so if you can apply binomial theorem on this it will start becoming one i plus a plus a square plus a cube and so on all the way till infinity okay now the moment you are given that a cube is a null matrix you'll start realizing that all these terms will start vanishing off because if a cube is null even a four will be null even a five will be null and so on and so forth isn't it right so ultimately you will be left with these three terms only correct so f of a would be i plus a times i plus a plus a square right let me multiply so it's i into i i into a square that's going to be repeated then a into i then a square then a cube okay so finally it becomes and again a cube is going to be null again so it's going to be a plus two a square so option number b is correct okay so none of you were correct in this case yeah option b is correct now we'll talk in general about the matrix polynomial that we have just now discussed okay many a times what happens is you would be given a polynomial something like f of x equal to this would be some polynomial okay a0 a1 x a2 x square and so on okay let's say you have been given a polynomial like this then they would ask you find f a where a is some matrix okay where a is some square matrix see what they mean to say is that you have to find a not i so where there is a one it will become a i there i of the same order as the matrix a itself let's say a is a square matrix of order n okay then this one will become i n of a1 i sorry a1 a sure sure as we can we can have so it's a1 a square okay a3 a cube and so on till a n a to the power of n is that fine a lot of questions based on matrix polynomial has been asked let us take a question that has come in one of the entrance exams let's take this question a b okay so let a be this 3 by 3 matrix and this satisfies x to the power n or you can say f of x is equal to x to the power n this satisfies let me write it like this this satisfies x to the power n is equal to x to the power n minus 2 plus x square minus 1 polynomial equation n is a value greater than equal to 3 is that fine now consider that there are three matrices u1 u2 u3 which are column matrices so these are u1 u2 u3 are column matrices or you can say they are columns rather than saying column matrices they are columns of u which is a 3 by 3 matrix okay such that such that to the power 50 u1 is 125 25 column matrix a to the power 50 u2 is 0 1 0 and a to the power 50 u3 is 001 okay question number one determinant of a to the power 50 second trace of a to the power 50 and third one determinant of u itself okay the question is clear i'll repeat once again what does the question talk about it says that there is a 3 by 3 matrix a okay which satisfies this polynomial equation x to the power n is equal to x to the power n minus 2 plus x square minus 1 n greater than equal to 3 and there is u1 u2 u3 are the columns of a matrix u which is a 3 by 3 matrix such that a to the power 50 u1 is this a to the power 50 u2 is this a to the power 50 u3 is this find determinant of a to the power 50 first of all second trace of a to the power 50 and finally determinant of u yeah that is pretty simple because this you can always find out by using the formula determinant a to the power 50 okay so i'm sure you would have found out by expansion about the first row so it's going to be one time zero minus one raised to the power 50 so that's going to be one undoubtedly what about trace of a to the power 50 okay so basically people are saying one for options one for question number three this is a question so Keith i hope you can see the question those who are saying determinant of u is equal to one that is also correct one three one are the answers absolutely correct what about others only few of you are responding Niranjan Ashutosh Sushant where are you guys okay let's discuss this now there are several ways in which i can solve it let's say i start with this matrix a and since this matrix satisfies x to the power n is equal to x to the power n minus two plus x square minus one so i can say a to the power n is equal to a to the power n minus two plus a square minus i okay now let's start putting the value of n as 50 first okay so when you put n as 50 you get a to the power 50 as a to the power 48 a square minus one when you put a to the when you put n as let's say 46 you get a to the power sorry 48 let's say a to the power 48 is a to the power 46 plus a square minus i okay let's say i put n as 46 now so a to the power 46 is equal to a to the power 44 plus a square minus i okay keep on doing it till you reach a to the power four which is a square plus a square minus i okay add them up if you add them you realize that this 48 this 48 cancels this 46 this 46 cancels this 44 will cancel with something down below this will also cancel with something up below okay so all together you will have how many a squares definitely you will have 24 eyes right from 4 to 50 from 4 to 50 how many eyes will be there how many eyes will i get over here so basically 4 6 8 all the way till you're going 50 so how many numbers are there so just simple 50 minus 4 by 2 plus 1 correct that's going to be around 23 plus 1 24 eyes i will get now a squares will be one more than that so this will be 25 a squares correct so a to the power 50 is as good as saying a to the power 50 is as good as saying 25 a square plus 24 right correct now finding a square is very easy a square is a into a you know your a values one zero zero one zero one zero okay so let's multiply here quickly so when you do the multiplication it becomes one it becomes a zero again a zero again a one again a one zero one zero one correct so a to the power 50 would be 25 a square that is 25 00 25 25 0 25 0 25 25 plus 24 i 24 i would be this but minus 24 i not plus 24 minus 24 so there'll be a minus here yeah thanks for correcting so this will become a one zero zero then 25 one zero 25 zero one but this clearly will answer the first two questions for sure what is a trace trace is going to be the sum of this so trace of a to the power 50 is going to be three okay determinant also can be found out from here also you can expand it with respect to the first row that will be one also okay next is you want to find out what is your u yes or no and it has been given that it has been given that a to the power 50 u 1 is 125 25 so let's say u 1 is made up of x y z so x y z multiplied with 1 0 0 25 1 0 25 0 1 this is going to be 1 25 25 okay which clearly implies x is 1 okay 25 x plus y is 25 and 25 x plus z is also 25 which means x is 1 y is 0 z is also 0 correct so u is made up of u 1 which is 1 0 0 in a similar way you will also found out that yes somebody is rightly mentioning that it's going to be the identity element is going to be an identity correct so determinant of u will also be one this is one approach of solving the problem this approach is clear to all of you now I'll show you another approach that is by the use of something called kelly-hamilton theorem all of you please listen to this I think I'd already discussed this in the videos but I'm going to repeat it yet again a kelly-hamilton theorem says that every square matrix every square matrix satisfies its characteristic equation satisfies its characteristic equation now what is the characteristic equation let us talk about this first a minus delta i where delta is some or you can say x i where x is some variable if you find the determinant and equate it to 0 this equation is called as the characteristic equation okay yeah absolutely absolutely so when we talk about the previous question where my determinant where my matrix a was 1 0 0 1 0 1 and 0 1 0 let's do subtraction with x i i is the identity matrix of order 3 and let's find the determinant of this and equate it to 0 so by the way this will be 1 minus x 0 0 1 minus x 1 0 1 minus x okay put this to 0 let us expand this what do we get we get 1 minus x times x square minus 1 correct and this is equal to 0 correct let us expand this one so it becomes x square minus 1 minus x cube plus x equal to 0 so which is x cube minus x square minus x plus 1 is equal to 0 okay so this equation says the kelly-hamilton theorem says that even the matrix a will satisfy the same equation so if you do this you'll always get a null you can check this out if you want okay so a will satisfy this characteristic equation yeah i'll come to those eigenvalues and all sukih okay is that fine is it clear with everyone oh something very interesting here if you see this equation let's say its roots are what will be the roots of this 1 1 and minus 1 correct yes or no now the sum of the roots the sum of the roots will actually be the trace of a check it out so if you see your a matrix just add the elements in the diagonal position you will get 1 as the answer right so trace of a is 1 and that will actually match with the sum of the roots so if you add them up they'll also get a 1 it always happens okay product of the roots is going to be the determinant of a so if you check this out the product of this if you multiply them all you will get a minus 1 right and try finding the determinant of a determinant of a if you find we had already seen you'll get a minus 1 not only this there are more interesting properties the property is if let's say I call these roots as lambda 1 lambda 2 lambda 3 okay so if lambda 1 lambda 2 lambda 3 are the roots of are the roots of the characteristic equation of a by the way these roots are officially called the eigenvalues so I think somebody was mentioning eigenvalues who was it yeah sukih it was mentioning the roots of the characteristic equation are called the eigenvalues okay you'll realize the importance of eigenvalues when we talk about eigenvectors as of now is just a shortcut for you to solve many polynomial based questions on matrices so listen to this what I have to say further on this if lambda 1 lambda 2 lambda 3 are the characteristic equation are the roots of the characteristic equation of a then if you if you take a to the power n then lambda to the power n lambda 2 to the power n lambda 3 to the power n would be the eigenvalues of the characteristic equation of a to the power n are you getting this point so little while ago I was talking about a to the power 50 correct so the eigenvalues or you can say the roots of the characteristic equation for a to the power 50 would be just raise these numbers to the power 50 1 to the power 50 1 to the power 50 minus 1 to the power 50 so that's 111 again so if I have to tell if I have to find trace of a to the power 50 it is going to be the sum of these roots right so that's going to be 1 plus 1 plus 1 that's going to be 3 are you getting it so I don't have to really calculate a to the power 50 to know these values are you getting this point okay but of course the last part of the question made it necessary for us to get a okay there are more characteristics for example if you want to find out the roots of if you want to find out the roots of characteristic equation of a inverse it will be 1 by lambda 1 1 by lambda 2 and 1 by lambda 3 right so the same theory is true even for this is that fine so I hope you have a new perspective about solving questions which are involving matrix polynomials so let's have a break night now let's take a break on the other side of the break we'll talk more about solving questions so let's take a break I will see at 6 let's say 13 p.m. let's meet at 6 13 p.m. all right welcome back hope you guys are back so based on the last concept that we had like mathematics polynomials let's take a question I'm sure most of you would have done the basics of integration right everybody's familiar with basics of integration okay so there's a question that I have based on that let's say if e to the power a is defined as i plus a plus a square by two factorial a cube by three factorial and so on and this entire expression as a matrix is half of f of x g of x g of x f of x a is a matrix which is given by x x x x for x lying between 0 and 1 okay and of course you know what is i i is an identity matrix of order 2 so i is basically an identity matrix of order 2 question is find the integral f of x by g of x options are ln e to the power x plus e to the power minus x plus c ln e to the power x minus e to the power minus x plus c e to the power 2x minus 1 plus c and option d is no doubt yes guys any idea you have to do this operation you have to do this entire operation and equate it to a matrix like this find what is your f of x in g of x no b is not correct sorry astosh b is not correct okay see it's not a rocket science here if we talk about a square first of all what is a square let's see the pattern of course we cannot sit and calculate till infinity but at least we can figure out the pattern right the pattern is if you multiply the rows with the columns it becomes x square plus x square which is 2x square again it is 2x square 2x square 2x square correct if you do a cube i'm sure it will become 2 square x square 2 square x square 2 square x square 2 square x square correct so it keeps on going like this correct now when you're finally doing e to the power a as i plus a plus a square by 2 factorial a cube by 3 factorial and so on i just have to pick up the first row elements like one from this x from this then 2x square by 2 factorial correct then you have 2 square x cube i'm sorry this is x cube x cube by 3 factorial and so on right okay second element would just have x 2x square by 2 factorial 2 square x cube by 3 factorial and so on right third one will have x 2x square by 2 factorial 2 square x cube by 3 factorial the same stuff that we have over here will be repeated over here and this will again have one extra one will be coming extra because of the one coming from this i let's do one thing from this very first element of this e to the power a matrix let me take a half common okay so inside i'll get a let's say 1 okay let's say 2x and this would become 2 square x square by 2 factorial 2 cube x cube by 3 factorial and so on and at the end i can add a half okay because one is made up of a half plus a half correct rest all of this all all the elements are adjusted right in a similar way here i can say half times 1 plus x plus again 2 square x square by 2 factorial da da da da and i just subtract a half because half was not there in the question okay and i'll repeat the same over here as well i'll repeat the same stuff over here as well oh by the way 2x will come okay and this stuff will be repeated over here if you see this clearly it's actually half e to the power 2x this is actually e to the power 2x expansion correct plus half this term is half e to the power 2x minus half expansion and this term here is again half e to the power 2x minus half this again half e to the power 2x plus half take the half common take the half common but becomes e to the power 2x plus 1 this becomes e to the power 2 x minus 1 Again e to the power 2x minus 1 e to the power 2x plus 1 now you know what is your f of x this this is your f of x and this is your g of x. So ultimately the entire process was to find out my f of x and g of x that means ultimately I have to do this integration. What was the question by the way let me just go back. I think question was f of x, g of x by f of x correct. G of x by f of x. Now how do we do this? Multiply and divide with e to the power minus x. So it will become e to the power x minus e to the power minus x by e to the power x plus e to the power minus x dx. Now let e to the power x e to the power minus x be t. So e to the power x minus e to the power minus x dx will become dt. Just as good as integration of dt by t which is ln mod t plus c. You don't need a mod because the term here is already a positive term. So it becomes ln e to the power x plus e to the power minus x plus c which I think is option number. Let me check out which option number was that. I think that was option number A. If it is f of x by this then it will become e to the power x plus e to the power minus x by e to the power x minus e to the power minus x dx. Then you have to take this as t then e to the power x plus e to the power minus x dx will become your dt. So it will become ln of mod e to the power x minus e to the power minus x plus c. So option b is correct. I think Ashutosh told b but he changed his answer I think. Oh he started with b then he changed his answer. Yeah, I remember you said b. Is that fine? So one last question we will take up on this concept. If A is 2 1 1 3 sorry 2 3 4 and minus 1 minus 1 minus 2. Okay, find trace of inverse of A cube. Oh even Sukir said that. No but he didn't say any answer. Yeah, so what is the best way to solve these kind of problems? Will you really sit and find A cube? Will you really sit and find A cube for such kind of problems? No, right? We will use Cayley-Emmelden theorem. Yeah, eigenvalues concept. Very good. So first let us find out the characteristic equation. So I will directly write down. Okay, put it to 0. Let's expand with respect to the first row. So that's going to be x minus 3 times x plus 2 plus 4 minus 1. So it becomes negative of 2 plus x times 2 plus 4 and 1 is minus 2 plus 3 minus x. Are you not following Cayley-Emmelden theorem? That's going to give you answer much faster. Okay, let's check it out. So 2 minus x. Okay, and this is going to be x square minus x minus 6 is minus 2. And this is going to be 4 plus 2x and a minus 4. So this will go off. We raise this off. And on this side, I will going to have 1 minus x. So let's check what do we get from here. So this becomes x plus 1 simply. And this becomes 2x square minus 2x minus 4 minus x cube plus x square plus 2x. So that's going to be minus x cube and 2x square plus 1 will be 3x square. And this and this gets cancelled. So you have plus x minus 3. Okay. So if you put this to 0, if you put this to 0, you'll get the characteristic equation. And I'm sure it's pretty much factorizable if you pull out a minus x square common as x minus 3 plus 1. So x square minus 1 times x minus 3 equal to 0. So x can be 1, 1 and 3. Sorry, 1 minus 1 and 3. Right. Now, having found this, if you have to find trace of a cube, okay, that means first do 1 by lambda cube of all the eigenvalues you have got to summation lambda 1 by lambda cube. So basically do 1 by 1 cube, 1 by minus 1 cube and 1 by 3 cube. So this and this is going to cancel. So answer is going to be 1 upon 27. Is that fine? So Cayley Hamilton theorem makes our life so, so easy. So if you know your lambdas of A, you know that for A to the power n, it would become lambda to the power n. Every eigenvalues or every root will be raised to the same power. Okay. And trace is nothing but the sum of all these eigenvalues for that particular matrix that you are looking for. Is that fine? Had I asked you what is the trace of A to the power 5, what would you do? You'll do 1 to the power 5 minus 1 to the power 5 plus 3 to the power 5. That's 243. Isn't that correct? Yes or no? Great. Now I'll talk about adjoint of a matrix. So we were waiting for adjoint of a matrix. This is symbolized as ADG of A. First of all, adjoint can only be found for a square matrix. Okay. How do you find adjoint of any matrix? Adjoint of a matrix is nothing but it's the cofactor matrix of A transpose. Again, I discussed this in my video, but I'm going to repeat this once again. What is cofactor matrix? Cofactor matrix is a matrix formed by the cofactors of every element present in the matrix. Let me take an example. Let's say A is 1, 2, 3, minus 5. Okay. So if I have to find cofactor of A, what I'll do is I'll first find cofactor of 1. By the way, cofactor of any element, let's say cofactor of 1 is nothing but minus 1 to the power rho plus column position of 1 times the number that you see when you hide these two. So if you hide the rho and the column where 1 falls, whatever number you see, that number will come over here. So that is going to be minus 5. So here, this would be minus 5. Yeah, I've already done this and I'm sure people would know this. Just for the benefit of everybody, I'll repeat this, cofactor of 2 would be minus 1 to the power 1 plus 2. Why 1 plus 2? Because 2 comes from the first or second column position. Hide this, hide this, you get a 3. So this becomes a minus 3. Okay. Cofactor of 3 would be minus 2. Cofactor of minus 5 would be 1. Okay. If you have to find the adjoint, you just transpose this matrix A. You just transpose this cofactor matrix of A. So it becomes minus 5, minus 3, minus 2, 1. Okay. Ideally speaking, what has happened? For a 2 by 2 matrix, there's a simple rule to find the adjoint. So if you have A, B, C, D, the adjoint of a 2 by 2 matrix is very simple. You switch the position of A and D. So as you can see here, 1 and minus 5 position have got switched. So you switch the position of A and D and reverse the sign of the elements in the other diagonal. So these two signs are reversed and these two positions are swapped. Is that fine? Similarly, you have a shortcut method for finding, shortcut method for finding adjoint of A where A is of order 3 by 3. Okay. I'll explain this shortcut method by an example because in a 3 by 3 matrix, you will have 9 elements. And finding the cofactors for all those 9 elements may be, may take a lot of time. So it will be a time consuming process for you. So there is a shortcut method for it. Let me take an example to illustrate this. I just follow this example. Let's say I want to find adjoint of 1, 2, minus 1, 0, 3, 4, 2, minus 1, 6. Okay. Let's say I want to find adjoint of A. Now all of you please listen to this method. What I do in this case is, first we write down the elements of this matrix as it is. So basically I copied everything. Okay. Then copy the first row after the last row. So 1, 2, minus 1 is copied. Then copy the second row after this one. So basically copy these two once again. Okay. Then copy this column over here 1, 0, 2, 1, 0. Copy this column over here 2, 3, minus 1, 2, 3. Okay. I am slightly going fast because you have already done this in the video shared with you. Okay. Now what you do is erase this row, erase this column. Okay. Now start cross multiplying as you would do to find a 2 by 2 determinant. Okay. So 18 minus minus 4 will be going to be 22. So right here. Okay. Then 8 minus 0, 8, 0 minus 6, minus 6, 1 minus 12, minus 11, 6 plus 2, 8, 4 plus 1, 5, 8 plus 3, 11, 0 minus 4, 3. Okay. This is the cofactor. This is the cofactor of it. Okay. So if you know the cofactor, you can easily find out the adjoint. Adjoint would be nothing but the transpose of it. So this is going to be your answer. This is going to be your answer. Is that fine? Yes, yes. We are trying to make it cyclic. Is it clear? I would like you to try out. All of you please try out. Find adjoint of A where A is 1, 3, 6, 4, 2, 1, 0, 1, 5. If done, let us discuss this. So quickly we will do it this time. That is why it is, there is no point of calling it as a shortcut if you are taking a lot of time doing it. First copy this, then copy the first row, then the second row. Copy the first column, whatever you have obtained so far. Copy the second column, whatever we have obtained so far. Okay. Now delete this. Delete this. Cross. So 10 minus 1 will be 9. 0 minus 20 will be minus 20. I am directly transposing it so that we do not have to waste time transposing it. 4 minus 0 will be 4. 6 minus 15 would be minus 9. 5 minus 0 would be 5. This would be minus 1. This would be 3 minus 12 which is minus 9. This will be 24 minus 1 which is 23. 2 minus 12 which is minus 10. Is this the answer that you got? This would be your answer. Okay. So I transpose it while I was multiplying it so I saved my time. Okay. Now what is so great about this adjoint? Why all of a sudden we are studying adjoint? What is the advantage of adjoint? See there are certain properties that adjoints follow. The very first and the important is a into adjoint of a is nothing but adjoint of a into a that means they are commutative and always gives you a scalar matrix. By the way, scalar matrix is a matrix whose diagonal, leading diagonal is k which is a nonzero scalar quantity and all other other elements should be 0. So something like this is a scalar matrix 2 2 2 0 0 0 0 0 0. So in case of exactly got up it actually helps us to find out the inverse. So this property is the most important property which actually helps us in finding inverse of it. Okay. I will talk about how to find inverse of it by the use of this property but right now it is very important to understand that this is actually a scalar matrix. Okay. Scalar matrix whose diagonal positions would be nothing but the determinant of the matrix A. Okay. For example, if I take determinant A to B let's say 10 then doing this would give you a matrix let's say n is 3 then doing such activity like A into adjoint of A would give you a matrix like this 10 10 10 0 0 0 0 0 0 0. Is that fine? Question is how? How it always gives me this matrix? It's very simple. All of you try to just understand this. Let's say A is made up of A, B, C, D, E, F. I'm just taking an example of a 3 by 3 G, H, I. Okay. Now what is the adjoint matrix? It's basically nothing but cofactor of A, cofactor of B, cofactor of C, cofactor of D, cofactor of E, cofactor of F. Cofactor of G, cofactor of H, cofactor of I. Correct. When you do this multiplication that means row with column. Okay. The first so first column element is A, C, A plus B, C, B plus C, C, C. Correct. Okay. Next element would be A, C, D, B, C, E and C, C, F. Right. If you keep on doing it, you would realize that only in the diagonal positions. Let me write the diagonal positions. Rest whatever elements are coming, I'm not going to write it. Now these terms itself is called the determinant. Okay. So this is the determinant, this is the determinant, this is the determinant. While all these terms would be 0, 0, 0, 0, 0. Okay. 0. This is the property of a determinant which says that you will soon study this property when you're doing determinants. Determinant is what? Determinant of a matrix is nothing but if you multiply, let's say the first row elements with its respective cofactors. Okay. I equal to 1 till n. This is what gives us the determinant. But if you do something like this, first row elements multiplied with, let's say the cofactors of the second row, I equal to 1 to n, you'll always get a 0 as the answer. This is a very important property. So determinant is only obtained when you multiply the elements of a row with the respective cofactors of that particular elements and add. But if you multiply the elements of a row with the respective cofactors of some other row elements, you'll always get a 0. Okay. That's how this property actually comes up. So you get a scalar matrix over here whose diagonal positions will be the determinant of the matrix A. Is that fine? Okay. So there are other properties also, I'll quickly talk about it. Property number two. So a joint of A, B follows the law of reversal and joint of B into a joint of A. Next property is determinant of a joint of A is basically determinant A to the power of n minus 1. This can come for your school as well. School questions can come on this. Can you prove this? You prove this. Determinant of a joint of A is determinant A to the power n minus 1, where n is the order of A. n is the order of the square matrix A. Done. Just write down done on the screen on your chat box. So basically again, go back to the same property which we discussed, A into a joint of A is nothing but a scalar matrix like this, correct? Let's take the determinant on both the sides. This will become determinant A into determinant a joint of A and this will become determinant A to the power n into 1, correct? It's because if you have a scalar matrix and you're finding the determinant, let's say K, K, K, 0, 0, 0, 0, 0, 0, it just becomes the product of the elements in the diagonal position. So this is basically KQ, right? You can actually generalize this. If it is a diagonal element, A, B, C, the determinant of it will be A, B, C. So let's cancel out one of the factors. So basically it becomes determinant of a joint of A is determinant A to the power n minus 1. Is that fine? Okay, prove that a joint of a joint of A is this. Okay, let's discuss this. So again, going back to the same property, A into a joint of A is nothing but determinant A into i n, correct? Now do one thing, replace A with a joint of A, okay? So when you do that, it becomes a joint of A into a joint of a joint of A, right? This becomes determinant of a joint of A i n, correct? Now let us multiply with A, okay? So A here, A here. So A into a joint of A, again by this property will become determinant A i n into a joint of a joint of A, okay? And this will become determinant of a joint of A into A, correct? Now i n into any matrix will become the same matrix. So you can just drop this i n, this i n is of no use, okay? And this term is determinant A to the power n minus 1. We have already seen that. And on the left hand side, we get this. Divide by determinant A both sides. You get a joint of a joint of A as determinant A to the power n minus 2 into A, right? Hence proved. Is that fine? One last property, a joint of A transpose is same as a joint of A whole transpose, fine? So guys, we will stop here. Next class, when we meet face to face, I will be carrying with the questions on a joint and we will take over to inverse and then elementary transformations and how to find inverses to elementary transformations. And that will be followed up by solving of system of linear equations by matrix method. So I think one more complete class would be required for me to wrap up matrices, okay? So the class is still 7 Ashutosh, so we are ending it up now, okay? Thank you everyone for joining in. Bye-bye. Good night. See you.