 Hi and welcome to the session. Today we will learn about multiplication of matrices. Suppose we are given a matrix A given by aij of order myn and another matrix v given by bjk of order n by p then the product of the two matrices A and v will be another matrix c given by cik and it will be of order m by p where the element cik is sigma aij into bjk where j goes from 1 to m. Now you need to remember one thing that the product of two matrices A and v is defined if the number of columns of A is equal to the number of rows of the matrix b. For example here the order of matrix A is m by n and the order of matrix b is n by p. Now as we can see that the number of columns of A is n and also the number of rows of b is n so that means they are equal and thus the product AB is defined. Also the order of the product matrix AB will be m by p. Now to make it more clear let's take one example of multiplication of matrices. Here we are given two matrices A and v and we need to find the product matrix c. Now as we can see that the order of matrix A is 2 by 3 and the order of matrix B is 3 by 2. Now here number of columns of matrix A is equal to the number of rows of matrix B that means the product AB is defined and the order of the product matrix C will be 2 by 2. So it will be a square matrix of order 2. Suppose the matrix C is given by c11, c12, c21 and c22. These are the elements of matrix C. So here first of all let's find out c11. Now for the element c11 we will take first row of matrix A and first column of matrix B. Now we will take each element of this row and we will multiply it with the corresponding element of this column and we will add their product. So here we will take 3 into 2 plus minus 1 into 1 plus 3 into 3. This will be equal to 14. Now let's find c12 for this. We will take first row of matrix A and second column of matrix B. Now let us multiply each element of this row with the corresponding element of this column. So this will be equal to 3 into minus 3 plus minus 1 into 0 plus 3 into 1 and this will be equal to minus 6. Now let's find out c21. For this we will take the second row of matrix A and first column of matrix B. Now let's multiply the elements. This will be equal to minus 1 into 2 plus 0 into 1 plus 2 into 3 and this will be equal to 4. Now we have to find c22. For this we will take second row of matrix A and second column of matrix B and we will multiply the corresponding elements. This will be equal to minus 1 into minus 3 plus 0 into 0 plus 2 into 1 and this will be equal to 5. So the matrix C will be 14 that is c11 then c12 that is minus 6 then 4 and 5. So this is the product matrix C which is equal to A into B. So I hope the multiplication of matrices must be clear to you. Now the multiplication of matrices is not commutative that is if we are given two matrices A and B then A B will not be equal to BA but multiplication of diagonal matrices of same order will be commutative. Now if we are given two numbers A and B such that A into B is equal to 0 then we can say that either A is equal to 0 or B equal to 0 but this is not true in the case of matrices. So if the product of two matrices is a zero matrix then it is not necessary one of the matrices is a zero matrix. For example for the given two matrices A and B the product A B is a zero matrix but as we can see either A is a zero matrix or B is a zero matrix. Now let's move on to the next topic that is properties multiplication matrices. Multiplication of matrices satisfy the following three properties. First property is the associative law. This states that for any three matrices A B and C A into B into C is equal to A into B into C. Whenever both sides of the equality are defined that is the product of these matrices is defined. So we can say that product of matrices is associative. Now the second law is the distributive law. This states that for any three matrices A B and C A into B plus C is equal to A into B plus A into C. Second A plus B into C is equal to A into C plus B into C. Wherever both sides of the equality defined and the third is the existence multiplicative identity. According to this property for every square matrix A there exist an identity matrix of same order in I such that I into A is equal to A into I is equal to A itself. So with this we have finished this session. Hope you must have enjoyed it. Goodbye, take care and have a nice day.