 Hello and welcome to the session. In this session we will discuss matrix multiplication. First of all we discuss the multiplication of a matrix by a scalar. Suppose we have a matrix A with elements A, B, C and D and if we have K be any real number or say a scalar then the product of the scalar or the real number K and the matrix A with elements A, B, C, D is obtained by multiplying each element of the matrix with the scalar K. So we will have a matrix with elements K A, K B, K C and K D. So consider a matrix say a matrix B with elements 1, 2, 3, 4 then 5B would be a matrix obtained by multiplying each element of the matrix B with 5. Since in this case we need to multiply the matrix B with the scalar 5. So we have a matrix with elements 5 into 1 is 5, 5 into 2 is 10, 5 into 3 is 15, 5 into 4 is 20. So this is the matrix 5B obtained by multiplying the scalar 5 by the matrix B. Next we discuss the multiplication of two matrices. Suppose we have two matrices A and B and we need to multiply these two matrices then we product AB of the two matrices A and B is obtained only if the number of columns of matrix A is equal to the number of rows of matrix B. So if given two matrices and we need to find the product so first of all we have to see that the number of columns of the first matrix is equal to the number of the second matrix to be multiplied with the first matrix and the product matrix AB has same number of rows the matrix A number of columns the matrix that is the product will have the same number of rows as the first matrix and the same number of columns as the second matrix. This means if order of the matrix A is M by P and order of the matrix B is P by N then the order of the product matrix that is AB would be M by N. Now from the order of the matrices A and B we can make out that the number of columns of matrix A is same as the number of rows of the matrix B. Now as the product AB is obtained when the number of columns of A is same as the number of rows of B so in the same way the product BA would be obtained when number of columns of B would be same as number of rows of A. In the product AB A is called the pre-factor, the post-factor. Now let's consider a matrix A that is a 2 by 2 matrix with elements A1, B1 in the first row, A2, B2 in the second row. Consider a matrix B again a 2 by 2 matrix with elements in the first row as C1, D1 in the second row as C2, B2. Now let's see how we can multiply these two matrices. Now the number of columns in the matrix A is 2 and the number of rows in the matrix B is 2 so this means both these matrices can be multiplied and since A is also a 2 by 2 matrix and B is also a 2 by 2 matrix so the product of the two matrices AB would also be a 2 by 2 matrix. Now the product of matrices A and B is denoted by AB and this is obtained by multiplying the two matrices. Multiplication of two matrices is described as row by column multiplication. Now in the multiplication of the two matrices our first step would be to multiply the first row by the first column. Now this is the first row with elements A1, B1. This is multiplied by the first column with elements C1, C2. So the entries of this first row are multiplied by the entries of this first column in order and these products will then be added that is we have A1, C1 plus B1, C2 and this is the entry in the first row, first column of the matrix AB that is of the product of the two matrices. Now in the next step we multiply the first row by the second column so this is the first row with elements A1, B1. This A1, B1 is multiplied by the second column with entries B1, B2. So these are also multiplied in the same way that is we multiply the entries of the first row with the entries of the second column in order and these products are added to obtain the entry in the first row second column of the product AB. Then in the next step we multiply the second row by the first column. Now the second row in matrix A is A2, B2 that is it has elements A2, B2. We multiply the second row with the first column in which the elements are C1, C2 so this would give us A2, C1 plus B2, C2. This gives the entry in the second row and first column of the matrix AB. Then in the next step we multiply the second row with the second column. Now the second row is the row with elements A2, B2 so we have the row containing the elements A2, B2 is multiplied with the second column with elements D1, D2. So this is the second column with elements D1, D2 so here we have A2, D1 plus B2, D2. This gives the entry in the second row and second column of the matrix AB. Let us now consider an example in which we are supposed to multiply the matrices A and B where the matrix A has elements 1, 2, 3, 4. Matrix B has elements 5, 4, 3, 1. Let us now find out the product AB of the two matrices A and B. Now as the matrix A is of order 2 by 2 and matrix B is also of order 2 by 4 so this means that matrix AB would also be of order 2 by 2. We first multiply the first row of the matrix A with the first column of the matrix B to get the element of the first row, first column of matrix AB. So 1 into 5 plus 2 into 3 so this is equal to 5 plus 6 which is 11 so here we have 11. Next we multiply the first row of the matrix A with the second column of the matrix B so we have 1 into 4 plus 2 into 1 so this is equal to 4 plus 2 which is 6. So 6 is the entry in the first row and second column of the matrix AB. Next we consider the second row of the matrix A and we would multiply this with the first column of the matrix B so 3 into 5 plus 4 into 3 and this is equal to 15 plus 12 which is 27. So 27 is the entry of the second row first column of matrix AB. Next we consider the second row of the matrix A and the second column of the matrix B and we multiply these so we have 3 into 4 plus 4 into 1 so this gives us 12 plus 4 which is 16. So 16 is the entry of the second row second column of matrix AB. So thus we have obtained the product of the two matrices A and B. So this completes the session. Hope you have understood the matrix multiplication.