 Okay, good evening everyone. I'll start with the class now. So today we are doing matrix and algebraic identity. So I'll start with matrix first and then I'll go to algebraic identity. So I have most two topics into it. So I'm starting with definition of matrix. So matrix is something which is an ordered rectangular area of number of functions. So what do I mean by this statement? So try to understand, let me go a little bit back into the history of evolution of matrix and why it was evolved. Actually matrix was evolved when we were attempting to solve linear equations. We all know how to solve linear equation by substitution method or elimination method. So what is the other method of solving linear equation? So when there was an concept evolution of any other method to solve linear equation, matrix was found as a result of that. So suppose I tell you that, try to understand. Suppose I tell you that there is a person A who has 15 pens, how can you represent this particular information in terms of matrix? So you draw this square bracket and you write like this. If somebody has 15 pens and five notebooks, so these are number of pens and these are number of notebooks. So what happens is you write 15 here and five here. So this is how the information can be represented and that is why I'm telling that matrix is nothing but an ordered rectangular array of numbers. So this is how in an array, a number or function is represented, the number or function. So each individual element over, entries over here would be known as elements or entries. So whatever you enter over here, all these individuals are known as elements. You can also call it entries. Now, where is matrix used? Though this is not important from the both perspectives that is used in cryptography and electronics spreads it. So matrix, I'll again repeat is an ordered rectangular array. And when I say rectangular, it doesn't mean it will always be rectangular when I discuss type of matrix. You'll see square matrix are also there and either it will be a square or rectangle. So this is how matrix is studied. I mean, this is the definition of matrix which says that it is an ordered rectangular array of number or functions. And basically it was evolved to study or to solve simple linear equations because we already had method and while discussing any other method, this particular method was evolved. Now let me go to order of matrix. So try to understand what do I mean by order of matrix? So order of matrix is something like this. So matrix may have, so suppose how a matrix is represented. So a matrix is represented like this, a11, a12, this will not be there, a13 and likewise, a1m, a21, a22, a23. And I'll keep this as n, this is a1n, this is a2n. This is a1n, this is a2n and likewise, I'll have am1, am2, am3 and this will be amn. So when somebody asks you to represent matrix in a very generalistic format, you can represent it like this. Now order of matrix, as you can see it on over here, there are m rows over here. So one, two, three, four, like m rows over here and this is called column. So there are how many columns? There are n columns over here. So always remember the first one to be used over here is number of rows and the second one to be used over here is number of columns. So always remember this, there was a question which was given that in three into four kind of matrix, how many rows would be there? And I could see students making a lot of mistake in this. There were people who had written that there are four rows and not three rows. So always remember the first one is number of rows and the second one is number of column. And a lot of people make mistake in this. I have seen myself while teaching that whenever this kind of question is given like m into n or x into y and how many number of rows, how many number of columns, people do make mistakes. So do have a clear cut conceptual clarity on this kind of questions. That first one is number of rows and second one is number of column. So the order of the matrix is number of, it is pronounced like this. So if it is m into n, so it is pronounced as m by n matrix or m into n matrix or like here it is three by two matrix, three by three matrix, two by three matrix. So what does it tell two by three matrix? Means it will have two rows, one, two and it will have three columns. Now how many number of elements would be there? Number of elements is equal to, so if I tell you how number of elements, number of elements is equal to m into n. So total number of elements, total number of elements is equal to m into n, so which is number of rows into number of columns. So suppose I give you three by two matrix, three rows here and two columns here. So how many elements would be there? So a one, a two, a three, a four, a five, a six. Just by writing the matrix, you can see that there are six elements which is equal to three into two. So number of elements can be described by taking a number of rows and number of columns and then multiplying it all together. Now let me move to some other topic which is an example over here. So consider that you have following information regarding number of men and number of women. How do you represent it in three into two matrix? So first of all, what it is telling you, it is saying you that you have to represent it in three into two matrix. So it means that there would be three columns, sorry, three rows, so I'm writing r one, r two, r three and there would be two columns, so I'm writing c one and c two and I'm closing the bracket, I'm writing this is three into two matrix. Now what happens is you have been given this particular information. So there are number of men worker, 30 number of women worker are 25 in first company. So I'm writing 30 men here and 25 women here. Then in second company, there are 25 and 31. So I'm writing 25 and 31 here. And in the third company, the number is 26 and 27 and 26. So I'm writing 27 and 26. And this is how you make it. This particular thing is just to understand you, you don't need to write it. You should only write that the matrix of number of workers, of workers for the given information would be equal to and you should draw this particular matrix. So if you are drawing this particular matrix and you are giving this particular information, you will have clear cut understanding of what you have represented here and you can get full marks for this. Now let me move to other topic and there is also one or two question over here. So what I'm trying to do is that I am trying to give you a question. So if it is, if a matrix has eight elements, what are the possibilities possible orders it can have? So this question can be done by factorizing eight because what happens is eight has to be represented. Any matrix would be of order M into M. Now if a total elements is equal to number of rows into number of column, it means that total element is equal to eight and I know that M into N is equal to N. So what I'm doing, I'm breaking eight into two parts and by breaking eight into two parts, I'm trying to gain numbers which can be represented as two numbers whose multiplication can be represented as eight. So what we can do over here is we can write like this that M I can take as one and N I can take as eight. It means that there is only one row and there are eight columns. You can write numbers from A1 to A8 over here and it will be one into eight matrix. Now simply you can reverse the order. So you can reverse the order and you can write matrix like this. So there would be eight rows R1 to R2 and there can be only one column. So you can write numbers like A1 to A8. So this will be eight into one type of matrix. So what I'm doing here is I've taken one eight and eight one. Now you can take two and four. So what I'm doing here is I'm making two rows and I'm making four columns C1, C2, C3, C4 and I'm writing here what I'm writing here is I'm writing A1, A2, A3, A4 and similarly A5 to A8 to change it. I can just swap the number of rows and columns. So I can make four rows here from R1 to R4 and I can make two columns here C1 to C2 and what I can do is that I can write numbers from A1 to A4 and I can write numbers from A5 to A8. I'll have eight columns. Apart from this, see the number of elements can only be integers. So M and N can only be integers. So multiplication of two integers are eight. I can have only four possibilities like this. I cannot have more than these four possibilities. Why? Because if I need only integers multiplication of two integers is eight. These are the only four possibilities. So first, second example, the answer is 18, 8, 1, four, eight and two, four and four, two. So these are the four possibilities. Now let me go to the third question and third question is primarily based on application of what we have studied till now. So it tells me that we have to construct three into two metrics whose elements are like this. So what I'm doing over here is I'm finding out, I'm writing A ij is equal to one by two i minus three j. Now you understand the first i represents row and j represents column. Now to understand this question properly, it has been given that I have to construct three into two metrics. It means that number of rows has to be three and number of columns has to be two. So what I'll do over here is I'll write A 11, this will be A 12. This will be A 21, this will be A 22 and this will be A 31, this will be A 32. Now value of i and j, when you write metrics like this, value of i and j is automatically available to you. Like in A 11, i is equal to one and j is equal to one. Now let's put the value over here and this is the modulus sign, this is not the matrix sign. So what I do is i A 11 comes out to be one by two, one minus three into one, which gives me one by two, one minus three. So which is one by two into two that comes out to be one. So what I get over here is if I draw the same thing over here, A 11 comes out to be one. Now let's find out A 12. So to find out A 12, what happens over here is that you get one by two, one minus three into two. So this comes out to be five by two. So I write here five by two, this is not written, this coma is not there. Now A 21 would be equal to one by two, two minus three into one, which is equal to one by two only. So I write this as one by two. Now A 22 would be equal to, you look at here, A 22 would be equal to one by two, two minus three into two. So this comes out to be six, two minus six, which is four. So one by two into four will give me two. So this will come out to be two. Now let's go and find out A 31. So this is one by two, three minus three. So this comes out to be zero, so this is zero. And now I'll find out A 32. So this is nothing but one by two, three minus six. So this comes out to be three by two. So this is three by two. This is how you make metrics. So if somebody is giving you this kind of information and you have to form the metrics, the first thing that you need to do is form this generalized metrics because until and unless you have this generalized metrics available to you, the value of i and j would not be identifiable to you. And once you identify value of i and j, there is nothing else to be done other than some multiplication and addition subtraction. So this making of this generalized metrics help in case of solving the question correctly. So even if you don't draw it, and I will suggest that if this kind of question comes in your copy, in your answer sheet, you draw this generalized metrics. Only drawing this generalized metrics will give you at least one marks. If it is a three marks question, it'll make sure you get one marks question by drawing this generalized metrics. What happens in case of metrics question is that most of the time we don't make generalized metrics. And due to that, we tend to make a lot of mistakes. So I always suggest that whenever you are attempting a metrics question, even if you don't draw this generalized metrics on the notebook or on the answer sheet, better you draw it in the rough sheet so that you have basic idea of what you are doing because if it has been given three into two metrics and you take three as column and two as row, anyway, you tend to make that mistake which you should not do. So time and again, I'm focusing on the row and column thing and now I'm telling you that this kind of generalized metrics has to be made in most of the questions that you solve for metrics. If generalized metrics is available to you, what happens is most of the time you get the right answer. Now let me go to type of metrics and I made different kind of metrics here. So first one is column metrics. Now what do I mean by column metrics? So by column metrics, I mean that the first one which I'm taking over here is column metrics and by column metrics, I mean to say that that column metrics is like this. Column metrics will have number of columns equal to one it means that if generalistic type of metrics or order of generalistic metrics, I represent by M into N and if N is equal to one, so if it becomes M into one type of metrics, it becomes column metrics. That only one column is available here. So you can write it that if somebody is asking you definition of column metrics, first you should write that the metrics is said to be a column metrics. If it has only one column. So this is the generalistic description. Now you go and explain your generalistic description by giving the order of a general metrics M into N and you say that N has to be equal to one and if N is equal to one, now the order of generalistic metrics become M into one in that case and you can represent normally any metrics is represented in very short form like this A, I, J and I write the metrics M into N. Now for column metrics, this metrics you can write like A is equal to element A, I, J, M into one. If you write this, you will get the full marks. Somebody will understand that the person is understanding that the number of column has to be equal to one and he is doing that. So you will get the full marks for that. Now let me move to the row, row metrics. So in case of row metrics, you might have understood by the name itself. The row metrics is something where you have only one row. So if I have to define this, a metrics is, is said to be a row metrics if it has only one row. So now I have, I have written this. Now I'll define it like any general metrics, I'll take A matrix is A11, A21, A31 and I'll keep on going till the time I write AM1. So what happens over, sorry, I've made a column matrix here. In case of row metrics, I'll write something like this. So I'll write A11, A12, A13, till the time I write A1N. What I'm trying to do over here, any generalistic metrics would be written like that, AijM into N. If M is equal to one, so it becomes a row metrics. So I'll write it as A is equal to Aij1 into N. If you write it, you get full marks for this. Now let me go to square matrix. So just for the sake of definition, any square matrix, so let me write square matrix over here. So a matrix, if you should write like this, a matrix which has same number of, rows and columns. So that would be a square matrix. What I'm trying to say is that you can make something like this, you can write A11, A12 and A21 and A22. So this is two into two matrix. What I'm trying to basically say is that in N into N, it becomes a square matrix when M becomes equal to M. So you can write like this that any matrix A is AijM into N where M is equal to N, that matrix becomes your square matrix. Now let me go to rectangular matrix. So rectangular matrix means wherever M not equal to N. So if number of rows not equal to number of columns, so what do you get there? You get a rectangular matrix over there. So M is not equal to N. So I'm just writing like this, A is equal to AijM into N the order. So if M not equal to M, that is your rectangular matrix. Now let me go to diagonal matrix. What do you mean by diagonal matrix? So if in any square matrix, so first of all diagonal matrix is a type of square matrix. So first there would be a square matrix and in that there would be a diagonal matrix. So if in square matrix there is a diagonal matrix, diagonal matrix is something where elements are only available suppose I take three into three square matrix and so what will happen is there would be three diagonal elements this A11 suppose I'm writing A12, A21, A13, A21, A22, A23, A31, A32, A33. So this particular line is known as not this one. You should not take this opposite one. This particular line is known as diagonal and A11, A22 and A33 will not be equal to zero. It means that Aij where i is equal to j would not be equal to zero and Aij where i is not equal to j would be equal to zero. So let me make a diagonal matrix for you. I'm writing 2, 0, 0, 0, 3, 0, 0, 0, 5. You look at here the diagonals are not, 0, all other elements apart from the diagonals are 0. So this is what the diagonal matrix is. Now let me move to scalar matrix. Now what do I mean by scalar matrix? So a scalar matrix is a type of diagonal matrix. So what does it mean? If scalar matrix is, I'm just explaining it, scalar matrix is, so you should write definition like this, scalar matrix is a type of diagonal matrix where the diagonal elements are zero. Sorry, non, sorry, equal. So diagonal elements are equal. So any matrix which you take, suppose I take any two into two matrix. So it will be minus one, zero, zero, minus one or I can take three into three matrix. So I will make this as zero. This is three, this is three, this is three and all others are zero. So if in a diagonal matrix, the diagonal elements are equal that comes out to be a scalar matrix. Now, if you have to represent a scalar matrix in generalistic term, you see here how I'm doing it. Any matrix is represented by A ij m into n. So element I write where i is equal to j, elements are equal and where i not equal to j, element I mean A ij would be equal to zero. So that is how you have to write this. Now, let me move to identity matrix. So a square matrix in which elements in the diagonal are all one and rest all are zero is known as identity matrix. So what I'm trying to say is that, let's say a scalar matrix where diagonal terms are one. So let's take a two by two matrix and look at here diagonal term is one. This is known as identity matrix. So this can be represented as identity matrix. Now, let me go to zero matrix. Now, zero matrix means all the elements are zero. So if all the elements of a matrix is zero, then that matrix would be known as zero matrix. So you can represent it in normal term A ij m into m where you can write A ij is equal to zero where i is from one to m and j is from one to n. So you can write like this and you can say that this is zero matrix. So these were the type of matrix that exist here. Now let me move to another topic and that topic is transpose of the matrix and equality of the matrix. So what do you mean by transpose of the matrix? Transpose of the matrix means that exchanging rows with columns. So suppose I have, I'm taking a rectangular matrix. So suppose I have any matrix A as two, three, five and four to one. So this is a two into three kind of matrix. Now, if you exchange rows with columns, so this is your first row. First row would be row one would be made column one. So I'm writing column one here and I'll write all the elements of row one exactly in the column one. Similarly, this is row two. So row two would be converted to column two. So column two, all the matrix I will write like this. Now you should see that you should not start from the right hand side, you should start from the left hand side. So matrix starts from here. So I'm starting from two and writing two, three, five in a vertical manner so that it comes into a column. And similarly I'm taking four to one and writing it in a vertical manner in second column. So now what happens is now number of rows becomes two and number of columns becomes three and number of column becomes two. So here m into n is there if order of a matrix is m into n for transpose matrix, order will change to n into m because I'm swapping the number, swapping the number of rows and number of columns. So this is what I mean by transpose of a matrix. Now let me go to equality of a matrix and I have written a few conditions in the PPT itself that first of all, any matrix or if two matrices to be equal, they have to be of same order. So I am writing a matrix A which has an order m into n and a matrix B which has an order m into n, then only the two matrix A and B can be equal. If matrix B has x in, if of order x into y and where x not equal to m and y not equal to n or any one of them, then the matrix cannot be equal. So for matrix to be equal, it is very important or the most important criteria is they are to be of same order. The second condition is each element of A is equal to corresponding element of B. So if I make something like this A11, A12 and A1n and likewise, A m1, A m2, A mn and I'm making it equal to B11, B12, B1n and B m1, B m2 and B mn. So both the matrices are first of all m into n order. This is m into n order. This is m into n order. And now what I do is that if you have to make this to be equal, it means that Aij over here is equal to Bij. It means that corresponding terms are only equal. Corresponding elements are only equal. So you take element first row, first column here. You have to take element first row, first column here. Here you are taking element fifth row, fourth column. Here also you have to take element fifth row, fifth column. So it means that you cannot change that corresponding order of the element. It has to be same always. Now let me give, solve a question over here. So the question can be taken as, suppose I'm talking about equality. So let me take a question of equality. And suppose I give you a question like this where I talk about concept of equality. So I am taking a three by three matrix and where I have to find out few terms. So I'm writing those terms here. And those terms are, suppose this is three y minus two. This is two c plus two and this is zero. And I just close it. Now I'm asking what is x, y, z, a, b, c, whatever I have written here. So what I'll do is that this x plus three, this is first row, first column and these two matrices are equal. So x plus three would be equal to first row, first column of the second matrix. So I will write x plus three is equal to zero. I'll find that x is equal to minus three. Now this z plus four, which is first row, second column would be equal to first row, second column of the second matrix. So I can write that z plus four is equal to six. And I can say that z is equal to six minus four, this is equal to two. And two y minus seven is equal to three y minus two here. So I can write that minus y is equal to minus two plus seven, which is equal to five. So y will come out to be minus five. Now let me go to a. So here I see that a minus one is equal to three. Sorry, this is minus three. So a is equal to minus one, sorry, plus one minus three, which comes out to be minus two. Now I know that b minus three should be equal to two b plus four because this is third row, first column. I am comparing it with third row, first column here. So I get b minus two b is equal to seven, b comes out to be minus seven. Similarly, I can see here, second row, third column, that two c plus two is equal to zero. So c will come out to be minus one. So that's how you have to solve this question. So you should always remember that in case of equality, the first and the most important condition is that they have to be of same order. And the second important condition is that when you are making the elements equal, the corresponding elements are only equal. It has to be taken care of always. Now let me move to addition operation on matrices. So I'm starting with addition and subtraction. So it's the same thing again. So if order is not same, addition and subtraction cannot be performed. And what do you add and subtract? You add and subtract corresponding elements from both the sides. So I have written over here, there are two matrices. So I'm taking two into three matrix from two into three matrix A and B. So you see here, first row, first column is A11 and first row, first column is A11 here. So if I'm adding A and B, I'm adding first row, first column of both the matrix. So it comes out to be A11 plus B11. Now here it is A12 and B12. So I'm adding here A12 plus B12. And this is A13 plus B13. So this is A13 plus B13. So what I'm doing is I'm adding corresponding elements over here from both the sides. Now here only I will teach you scalar product of matrix. So what do I mean by that scalar product of matrix? So suppose I am writing a matrix, suppose I'm writing a two into two matrix. So suppose A is equal to three five minus two one. And I have to find out two A. So what do I mean by two A? I mean A plus A. So A plus A means three five minus two one plus three five minus two one. So I add three plus three six five plus five 10 minus two, minus two, minus four one plus one two. So actually I'm adding A two times. Rather than doing this, what I could have done that, if A is multiplied by two, so I could have multiplied each element with two, I could have got the same answer. So multiply each element with two, I'm writing here, you will get six 10 minus four and two. So a scalar product means whenever a matrix is multiplied by a constant, you multiply all the numbers, all the elements of the matrix with same constant. If a matrix is divided by a constant, you divide all the elements of the matrix with the same constant. So what happens over here is these kinds of questions do appear in the examination. So while multiplying or dividing, do take care that you are multiplying and dividing carefully because if it is a three into three or four into four matrix, you have to divide or multiply nine, nine elements or 16 elements. So it becomes, I mean, very calculation intensive and sometimes we tend to make calculation mistakes which we should avoid. So always be prepared when this kind of scalar multiplication or division comes, just take a little bit of caution while solving this so that you can avoid calculation mistakes. Now let me move to another topic which is multiplication of matrix. So what do I mean by or let's discuss the process of multiplication of matrix. So what happens? I have taken a three into three matrix here and I have taken a three into three matrix here. So if I generalize it, I take a 11, a 12, a 13, a 21, a 22, a 23, a 31, a 32, a 33, multiplied by b 11, b 12, b 13, b 21, b 22, b 23, and similarly 31, 32 and 33. Now multiplication method is pretty simple. You take first row and first column at the first time. So try to understand the order of the multiplication. What would be order of the multiplication? So I'm writing here if you have a matrix of m into n and you have a matrix of n into p here, suppose. So if you multiply these two matrix, you find order of the matrix as m into p. So first thing that you need to understand is that this number of column here and number of row here has to be same. If it is not same, then what happens? So that I'll discuss later how it is done. But first I'm taking this kind of multiplication. So see here I have taken three into three kind of multiplication. So what I'm doing is I am taking first row here and first column here. So I multiply a 11 into b 11, first term and first term. I take second term in the row and multiply second term in the column and add here. So a 12 into b 21. Plus I take third row in the third element in the row. So and here third element in the column. So a 13 into b 31. So this makes my first term of our first element of the multiplication matrix. So now what I do for the second term, I keep the row here, change the column here. So now the second term would be you look at here, I've taken a the first term here, multiply to the first term here, it becomes a j. Plus I take the second term here and second time in the column here, this becomes b m. I take the third time here and third time here, this becomes c p. Now what I have done over here for the second element is that this row remains same but the column over here changes. So it becomes a a first term here and first time in second column, so a k second term here and second time in the second column so this becomes b m and this becomes c q. Now again, the row will remain same and the column will change. So this I keep on doing. So AL, BO and CR all added together. Now I'm finished with all the rows here. Now I'll go to second, second row here. I take second row again. I'll multiply first here with first column, second with second column and third for third element with third column. Now I'm done with all the columns here. Now I'll go here. I take first element here, first element here, second element here, second element here, third element here, third element here. So it's actually row into column kind of multiplication is here in this case. So let me take a multiplication example and solve it for you guys. And I will also tell you, suppose A is 6, 9, 2, 3 and B is 2, 6, 0, 7, 9, 8. So if I have to multiply A into B, how will I do it? So you look at here, I'll take first column here, 6 into 2 plus I'll take second column here and I'll multiply to the 7. So this becomes 9 into 7. Now what I'll do? I'll take 6 here and multiply to second here. So 6 into 6 and plus 9 into 9. Then I'll take 6 into 0, so which becomes 0 and 9 into 8 which becomes 72. And I'll do the same thing here. I'm exhausted with all the columns here. Now I'm going to second row here, 2 into 2. Plus I take the second number here and multiply it to this, 3 into 7. Then I'll go to second column. So second row into second column. So 2 into 6 plus 3 into 9. And then I go to second row into third column. So 0 into 2 is 0 and 3 into 8. So that becomes 24. So this is nothing but 6 into 2, 12 plus 9 into 7, 63. So 12 plus 63 here. I have 36 plus 81 here and I have 72 here. Then I have 4 plus 21 here. I have 12 plus 27 here and I have 24 here. So it gives me 75 here. It also gives me 117 here. This becomes 72. This becomes 25. This becomes 39 and this becomes 24. Now let me come to order of the multiplication. Order of the multiplication is 2 into 3. Now what do I have here? Order of A is 2 into 2. Order of B is 2 into 3. Now I told you in the beginning itself that if you multiply 2 into 2 and 2 into 3, the column here and the row here vanishes. And what you get as a result is 2 into 3. So if m into n is multiplied by n into p, what you do is this n gets vanished and order of multiplication matrix would be m into p. So it can be easily asked to you that what kind of, I can give you a question that m into n and n into p kind of matrix is getting multiplied. What would be the order of resulted matrix? You should write m into p. So one more question can be there on this particular concept. So I have shown you how multiplication is done. Now what I'll do is, I'll take a few properties of multiplication. So let me take a few properties of multiplication. So properties of multiplication, first one it is non commutative. So what do I mean by non commutative? So non commutative means A into B is not equal to B into A. So why does it happen? Because in case of I had taken the example of, suppose I take an example of this. So I take one minus two, three and minus four, two, five. This is my matrix A. And I take an example of B. You take two, four, two, and three, five, one. It gives me a matrix of two into three and this is three into two. So A into B just you take, A into B is two into two matrix because this three gets vanished as I just told you. And B into A would be because this is three into two and this is two into three. So this two gets vanished. This would be three into three matrix. So what happens is here you have only four elements and here you have nine elements. So you can see that A into B and B into A. You don't need to multiply over here and check. You can do so and if you do so you will get that kind of matrix but just by taking the order of the matrix I have told you that M into N multiplied by N into P you get M into P kind of resultant matrix. So what happens over here is if I multiply two into three to three into two, two, three gets vanished. So you get two into two matrix and here you get three into three matrix, four elements here, nine elements here. So you cannot actually get the same matrix. So it is non commutative in nature and this is one of the most important properties of multiplication of matrices. So if you have this property in mind you can avoid a few mistakes. The second, two, three other properties that I would like to discuss is the associative law holds over here. So if a matrix is like this A into B and then multiplied to C this can be equal to A, B and C multiplied first and then it is getting multiplied. So associative law holds here. Then you can say that distributive law also holds here. So distributive law also holds here. So what do I mean by distributive law? So if this is the matrix A multiplied by B plus C you can write this as AB plus AC. This will hold true or if you take it from the other side so you can write A into B multiplied by C you can write it AC plus BC this also holds true. So this is what needs to be done. Now let me go to other properties of matrices. So these are a few properties of matrices you know when you can multiply the matrix when you cannot multiply the matrix I have already discussed transpose of the matrix and all those things with you. So matrix at class 10th level is only up to this level where you have to do multiplications you should do transpose and scalar multiplication because there are many questions which comes on the basis of scalar multiplication. So if you know scalar multiplication which I already discussed and there is nothing in the scalar multiplication other than multiplying or dividing all the elements depending on the question over there by the constant given over there. So you can solve most of the questions using the concepts that I have already discussed with you. Now let me move to the other topic which is algebraic expressions. Now algebraic expression is a very important topic for autonomous CBC board and you can see very tricky questions which come from this topic. So what happens is you mostly the question based on algebraic expression is simplification of algebraic expression which we call taking it into its simplest form. So most of the questions are based on finding the expression given in the simplest form. If that kind of question is not given then there would be few addition, multiplication, subtraction, division kind of question where multiple expressions would be given. They would be mostly correlated to each other and you will have to add, subtract, multiply or divide those expressions. So generally it is best on that concept. So this is not a very difficult topic but in rush of things, sometimes we tend to make mistake which I think that can be negated if we practice this kind of question a lot. So let me start with the discussing properties of integer first this topic. So first property of integer is pretty simple and I'm writing all the properties here. First property is sum of any number of integers will always be integer, will always be equal to an integer. The second property is that associative law and cumulative law. So I have already discussed associative law and cumulative law. So associative law is A plus B plus C and I put it in the bracket is equal to B plus C plus A. So this is your associative law and cumulative law tells me that A into B is equal to B into A. So associative law of multiplication is also true over here and in addition also commutative law is true. So multiplication is A into B multiplied by C is A into B into C that is also true. So these are the few properties. So all the properties hold to over here. Now let me go to properties of polynomials. Now what is properties of polynomials? So I will finish it in only two, three sentences. So any polynomial is first represented by Px or Qx. Qx it shows a polynomial in terms of x. Now I'm just saying I'm making a statement which is why these two things has been given together because most of the properties matches with each other. So I told that some of the two integers are always integer. Here I'm saying that some of the two or more polynomial will always be a polynomial. So here also associative law of addition and multiplication, commutative law, distributive law, all the laws that we have discussed will hold true for polynomial also. So if we have these things in mind, we can negotiate between the problems easily. Now we all know it so there is not much emphasis needed on this topic. Now let me go to another topic which is this. So analogy between rational number and rational expressions. So what is a rational number? Any rational number can be written as m by n. Similarly a rational expression can also be written as px by Qx. Now we all know that this n should not be equal to zero. Here also Qx should not be equal to zero. Now here also rational numbers and rational expression follow laws of in case of both addition and multiplication, it follows law of or I can say associative law, distributive law and commutative law. So all these laws are follows by rational numbers and rational expressions both. So now let me move to rational expressions property now algebraic expressions are basically study study of rational expressions. And any rational expression is represented by px divided by Qx where Qx is not equal to zero. Now this px by Qx need not to be a polynomial. You can see that Qx can be equal to one also here. Now take an example here. So suppose I have an example 3x minus four divided by 5x square plus 2x plus three. Now if somebody asks you whether this is a rational expression or not. So what will I say? I'll say that this is a rational expression because if I compare px, px is a linear equation and if I compare Qx, Qx is a quadratic equation. So px by Qx has to be a rational expression. Let me take another example for you. If I take this example, 5x to the power four root 3x cube plus 2x square minus five divided by 3x cube minus root 2x square minus nine. So if somebody asks me whether this is a rational expression or not, if I compare px, px is a biquadratic equation and Qx is a cubic equation. So px by Qx is a rational equation. But what about this which I'm going to write? I'm going to write this equation for you. X cube plus 5x square minus three divided by x square plus root x plus one. Now this polynomial is okay, but what about Qx? Qx is not a rational polynomial because here the power is not an integer. It is a square root which has been involved here. And wherever in variable integer is not there, the variable or like x to the power integer is not there. Those kind of situations, we will call that the expression is not a rational expression. So here as power of x is not integer, it is one by two. I'll see that this is not a rational expression. So that is how we have to identify which expression is a rational expression and which expression is not a rational expression. When you are looking at any algebraic expression, you have to look that in both numerator and denominator, what happens is the power of x has to be an integer. If it is a square root, cube root, or any other root, one to the power n kind of thing, it cannot be a rational expression. Now let me discuss something else which is rational expression in lowest form. Now how do you find any rational expression is in lowest form or not? So there is a very basic rule about finding any rational expression in lowest form and what is it? So you take any number. Suppose you take an expression like this, x minus three x plus five divided by x minus five. Now is this in lowest form or not? Or let me take another number three divided by seven. Whether this is in lowest form or not? Let me take a number two divided by six. Whether it is in lowest form or not? What do we do to check any particular rational expression or rational number is in lowest form or not? So we take numerator and we take denominator and we try to find out hcf of numerator and denominator. So greatest common denominator or hcf of numerator and denominator should be one. If hcf of numerator and denominator is one, it means that there is no common factor between numerator and denominator. So that common factor cannot be canceled out. The common factor is one. So you see here there is no common term between x minus three x plus five and x minus five. Hence the hcf over here is nothing but that is equal to one. Three and seven also hcf is one, but two and six hcf is not one. So this is not in its simplest term. So I know two and six hcf is two. So it means that two can be canceled out from here. So this can be written as one and this can be written as three. So if you divide numerator and denominator by hcf, it converts into simplest term. So that is what we have to take care of when we are finding out whether any particular expression, rational expression is in its simplest form or lowest form or not. So let me give you a question and the question is six x square minus five x plus one divided by nine x square plus 12x minus five. Now I need to know whether this is in simplest form or not. If it is not in simplest form, I need to find its lowest form or simplest form. So I am to find whether it is in lowest form or not. Here both numerator and denominators are quadratic equation. So what we need to do over here is we need to find out or we need to factorize the quadratic equation in both numerator and denominator. So let me do that for you. So here I can write six square minus three x minus two x plus one. And here I can write nine into five is 45 and 15 minus three is 45. So I can write that nine x square plus 15x minus three x minus five. So let me factorize it for you. This I can write at three x, two x minus one, minus one, two x minus one, this numerator. And denominator can be written as three x. So here you find three x minus five and here you can take minus one, three x plus five, sorry. And here you can write three x plus five. So what happens here is from numerator, here you can write three x minus one into two x minus one and denominator you can write three x plus five multiplied by three x minus one. Now in the numerator and denominator, you see that three x minus one is common. So what you can do over here is you can cancel out three x minus one. So this gives me two x minus one divided by three x plus five. This is the lowest form. So that is how you can solve these kinds of questions. So to identify whether any rational expression is in lowest term or not, we need to find out that the HCF of numerator and denominator is one or not. If HCF of denominator is one, it means that that is in simplest form or not. And then if we have, if it is not in simplest form, then that can be found in simplest term by canceling the factors properly. So now let me go to another, this thing. So I'll solve a few questions based on addition, subtraction. So let me solve a few questions on addition, subtraction. So if you look at the laws which are, which has been done, taken properly in case of addition, subtraction. So what happens over here is that suppose Px by Qx is one of your rational expression and Rx by Sx is second rational expression. So what you can do is that you can take LCM, Qx and Sx and this becomes Px into Sx and this becomes Rx into Qx. Now, similarly, in case of multiple subtraction, you can take subtraction here, this will turn out to be negative. So I'm writing positive, negative both here. Now, let me go to different properties of, so if I talk about multiplication, there are different properties involved with multiplications of, so first one is closure property. This tells me that if Px by Qx multiplied by Rx by Sx, numerators get multiplied together and denominators get multiplied together. So this is the first property. Associative law tells me that, it's associative law will hold true here, Px by Qx multiplied by Rx by Sx and I take Tx by Ux. So this can be written as Px by Qx multiplied by Rx by Sx and this can be written as Tx by Ux. So associative law holds true here. Let me go to cumulative law. So cumulative law also holds true here. So if I change this, just I'm changing this here only and I change this to, I change this to positive sign here for cumulative law. So what happens over here is, I will put a positive sign here and this will turn out to be, this is Px by Qx multiplied by Rx into Sx and plus you can write Px into Tx multiplied by Qx into Ux. This is what you can write in case of cumulative law. Now let me move to another topic. So let me take a few questions over here. In this case, we should solve a lot of questions. So let me take a few questions. The first question that I'm going to solve is over here is because there is nothing much except solving questions over here. So I am taking the first question over here and the first question tells me and this is a CBC question. This is x squared plus 8x plus 12 divided by x square minus 7x plus 12 multiplied by x square plus 4x minus 12 divided by x square minus 5x plus 4 and I have to write it in simplest form. So this can be written as x plus 6 because all are, because all are quadratic equations. So you can just factorize it. This you can write it as x plus 6x minus 2 and this you can write as x minus 4x minus 1. Now I'll check whether something is matching or not. So here in case of multiplication, nothing is matching. Now what I do is that if I change this multiplication sign with division sign, now let's see what happens. So if I change this multiplication sign with division sign here, so now let's see what happens. So I can write this as x plus 6x plus 2 divided by x minus 3x plus 4 multiplied by x minus 4x minus 1 divided by x plus 6x minus 1. So what I can do over here is this x plus 6 and x plus 6 is gone and once again this is x minus 2. So now what happens is in the numerator what I get is, in numerator what I get is I'll get x plus 2x plus 4x minus 1 and is there anything else cancelling out? So what cancels out over here is so and in denominator x minus 3x plus 4 and x minus 2. So you can write x plus 2x minus 4x minus 1 and then you can write x minus 3x plus 4 and x minus 2. This is what the answer would be. Now let me move to another question and then another question is, suppose I take this question for you guys. So let me take x cube minus 6x square plus 36x divided by x minus 7 and that is divided by x to the power 4 plus 216x, this is a good question and x square minus 4x minus 21. Now let me solve it one by one. So here numerator x can be taken as outside and this is x square minus 6x plus 36 and I can write the denominator like x by 7 and I can multiply it and I can write this x square minus 4x minus 27 is x minus 7x plus 3 and this is nothing but if you take x common this comes out to be x cube plus 216 and a cube plus b cube can be sorted out or can be written as a plus b a square minus a b plus b square. So I can write this as x plus 6x square plus 36 minus 6x. So you can write this as, so this goes in the numerator. So x minus 7x plus 3 goes in the numerator and what comes in the denominator is x into x plus 6x square plus 36 minus 6x. So this x minus 7x minus 7 is gone. This x and x is gone. This x square minus 6x plus 36 is gone. So what is left out in the numerator is x plus 3 and in the denominator what is left out is x plus 6. So this is the simplest form that we have been talking about and this is how it is converted into simplest form. Now let me solve another question for you. So I told that if in the beginning that if simplest form is not asked to you then what we can do over what we can do in that scenario or what kind of question can be asked in that scenario. So the first question that is in front of me is x square minus x minus 6 divided by x square minus 9 plus x square plus 2x minus 24 divided by x square minus x minus 12. So here in this question also what I will do is first I will do the factorization and then I will do the addition. So this numerator can be written as x minus 3x plus 2. Denominator can be written as x minus 3x plus 3. It is a square minus b square forms so a plus b a minus b where a is x and b is 3. And this is nothing but x plus 6x minus 4 and this is nothing but x minus 4x plus 3. So here x minus 4x minus 4 is gone and here x minus 3x minus 3 is gone. So what I get is x plus 2 divided by x plus 3 and here I get x plus 6 divided by x plus 3. Now as the denominator is same at both the places what I can get over here is I can get 2x plus 8 divided by x plus 3. This would be the answer. Now this is also not in our right answer to write because 2 is common here. So I'll take 2 common here. I write 2x plus 4 divided by x plus 3. This is my final answer. Now let me take another question for you. So I take another question which is 1 divided by x plus a plus 1 divided by x plus b. Plus 1 divided by x plus c plus ax divided by x square plus ax cube plus ax square plus bx x cube plus bx square plus cx x cube plus cx square. Now how do you solve this kind of question? So you see here you find similarity between this particular denominator and this particular denominator. So I'm solving it for one of them. I take 1 by x plus a and I'm taking ax plus x cube ax square. Now what I do is that I take out from this denominator I take out x square common. So I have ax in the numerator. I take ax is x square common. I have x plus a as the denominator here. I'll do this later. First let me show you for this. So what happens over here is this x and this x is gone. Now x, x plus a is my denominator. So here what I get is x and here what I get is a. So you see over here that this x plus a, x plus a is gone. What I get over here is one plus x. So similarly in this case also I can take one by x plus b plus bx divided by x. I'm taking common here only x square. x square I'm taking here common only and this becomes b and this x gone. So I can solve one by x plus b divided by x into x plus b and this can be written as x plus b divided by x multiplied by x plus b x plus b, x plus b gone. So I can write this also as one by x. This also can be written as one by x plus c and cx divided by x cube plus cx square can be written as one by x. So this answer would be three by x. This is your final answer. So if you have some questions like this you can you can solve it. Now I have another very good question in front of me which I would like to solve before I wrap up the session. So you will not get so it is something like this. x square minus y minus z square divided by x plus z square minus y square plus. So looking at this question I can know that what the pattern would be. Here x is there. So now y will come this y would be replaced by z this z would be replaced by x. So y square minus z minus x square divided by now this is x plus y whole square minus z square and plus this is nothing but z square minus x minus y square divided by y plus z square minus x square. Now how to do this kind of question? Now first look at the numerator. This is something of a square minus b square kind. So this can be written as a minus b a plus b. So I'm writing x minus y minus z. So this becomes minus and minus minus will become plus and the other one would be x plus y minus z divided by again this is a square minus b square format. So I can write it as x plus z minus y and x plus z plus y. So you see here this x plus z minus y and x plus z minus y is getting canceled out. So what I'm getting over here is x plus y minus z divided by x plus y plus z. So I should know that I will get similar kind of term over here so which will be y minus z plus x y plus z minus x divided by I get x plus y minus z and x plus y plus z. Now you look at this term this x plus y minus z is canceling this and what I get over here is y plus z minus x divided by x plus y plus z. Now this also can be written as z minus x plus y and it can be written as z minus z plus x minus y and here it can be written as y plus z minus x and here it can be written as y plus z plus x. So this and this x plus z minus z plus y minus x get canceled out and I get z plus x minus y divided by x plus y plus z. Now my denominator is same everywhere so I write x plus y plus z. Now you will see two x are positive here and here and one x is negative. So two x minus x is x, two y would be positive here and here and here it is negative. So two y minus y is y and two z would be positive here and here and here it is negative so I can write z. So x plus y plus z divided by x plus y plus z will turn into equivalent to one. So this is what the answer of this question is. Now I have another question for solving and this question is nothing but one divided by one minus x plus one divided by one plus x plus two divided by one plus x square plus four divided by one plus x to the power four. Now this kind of question I should know that see what happens over here. If I solve this question, this part only. So in the denominator I'll get one minus x square and what is waiting for me to be added is two divided by one plus x square where denominator is one plus x square. So one minus x square and one plus x square multiplied together will give me one minus x to the power four and I have one plus x to the power four. It means that something corresponding is waiting for me always to be added. So I first add this part together and this gives me a denominator of one minus x square. So this will be one plus x and this will be one minus x. So this comes out to be two divided by one minus x square and now what I'll add over here is two divided by one plus x square. So what happens is if I take two common this gives me one plus x square plus one minus x square divided by one minus x to the power four and again this x square, x square is gone. So what is left out is one plus one two two into two four, four divided by one minus x to the power four and which I add here. So four divided by one minus x to the power four four divided by one plus x to the power four. So I have one minus x to the power eight and I take four common here. So this becomes one plus x to the power four plus one minus x to the power four. This and this is gone. So four into two is how much? Eight divided by one minus x to the power eight. So the answer is eight divided by one minus x to the power eight and this is how this question can be solved. So looking at the pattern in this kind of questions is very, very important. If you are not able to find out pattern, generally if you look at all the questions which has appeared in your examination would be some kind of pattern. So some kind of pattern would be available in that question. It will be better if you look at the pattern and then try to solve the question. Like in the last question, looking at the denominator, I could identify the pattern. That the pattern was one minus x, one plus x which turns into one minus x square. Then one minus x square multiplied by one plus x square gives me one plus x to the power four which can be multiplied to one plus x to the power four and I can get similar values over there. So this is how you had to solve the question. Now look at here and another question to be solved is one divided by x minus two x minus three plus one divided by x minus two x minus one plus two divided by x minus three x minus five. Now what I do over here is I take x minus two common. So I take one divided by x minus two common. What left out with me is one divided by x minus three divided by x minus one. And I write it as it is over here. So this gives me one divided by x minus two and this is nothing but x minus one plus x minus three plus x minus three divided by x minus one multiplied by x minus three. Now let's solve it here. This is nothing but one divided by two x minus four is nothing but you take two common. This will be two multiplied by x minus two, x minus one, x minus three. So x minus two, x minus two gets cancelled over here and what I get from here is two divided by x minus one, x minus three. Now I'll add this to two divided by x minus three, x minus five. So here also you can see in the denominator x minus three and x minus three is equal. So I take two divided by x minus three common and what is left out is x minus one plus x minus five and this is equal to two divided by x minus three and this is x minus one plus x minus five divided by x minus one multiplied by x minus five. So this is equal to two divided by x minus three and this is two x minus six divided by x minus one, x minus five. So this is nothing but two multiplied by two taken from there and this is x minus three here and this x minus three, x minus one, x minus five will remain as it is, this and this gone. So my answer would be four divided by x minus one, x minus five, as simple as that and you cannot leave it like this. So you multiply it and then you write four divided by x square minus six x plus five. So this is your final answer. So this is what I had to teach in the session. So I am wrapping up the session by saying that both these topics are very scoring in match. So you should focus a lot on this. In algebraic expression, just try to look at the pattern which pattern needs to be followed, how the pattern is coming from solving the questions. I mean, if matrix is to be talked about in matrix, a lot of people tend to make a lot of numerical, I mean calculation mistakes, careless mistakes. Please be aware when you are multiplying two numbers, adding two numbers, subtracting two numbers because what happens is if your calculation is right, there is nothing which cannot be solved in these two topics. So especially matrix because you don't have anything apart from subtraction, addition and multiplication. So please focus on calculation errors. I mean, you should not make too many calculation errors. Also at the very same time, what happens is if you practice more questions with time those calculation errors will keep on going down. So practice a lot of matrix question, try to eliminate your calculation errors. In algebraic expressions, try to identify the pattern if you identify the pattern, you can solve the questions quickly. So that's all from me today. Thank you so much for attending this class and hope we'll meet soon. Thank you so much.