 Hello and welcome to the session. In this session we are going to discuss the following question and the question says that if A is equal to matrix containing elements 1, 3, 2, 4, B equal to the matrix containing elements 2, 1, 0, minus 3 and C equal to the matrix containing elements 1, 0, minus 1, 1 show that A into BC is equal to AB into C. Let us first start with the LHS which is A into BC. This is equal to the matrix containing elements 1, 3, 2, 4 into the matrix containing elements 2, 1, 0, minus 3 into the matrix containing elements 1, 0, minus 1, 1. This will be equal to matrix containing elements 1, 3, 2, 4 into a 2 by 2 matrix whose elements are the element in the first row first column is obtained by multiplying the first row of the first matrix by the first column of the second matrix that is 2 into 1 plus 0 into 0. The second element is the element in the first row second column that is 2 into minus 1 plus 0 into 1. Now the element in the second row first column is obtained on multiplying the second row of the first matrix by the first column of the second matrix that is 1 into 1 plus minus 3 into 0. Similarly the element in the second row second column will be 1 into minus 1 plus minus 3 into 1. This will be equal to matrix containing elements 1, 3, 2, 4 into the matrix containing elements 2 plus 0, 1 minus 0, minus 2 plus 0 and minus 1, minus 3 which is equal to the matrix containing elements 1, 3, 2, 4 into the matrix containing elements 2, 1, minus 2, minus 4. Now again when we multiply these two matrices we obtain a 2 by 2 matrix whose elements are the element in the first row first column is 1 into 2 plus 2 into 1. The element in the first row second column is 1 into minus 2 plus 2 into minus 4. The element in the second row first column is 3 into 2 plus 4 into 1 and the element in the second row second column is 3 into minus 2 plus 4 into minus 4. This is equal to the matrix containing elements 2 plus 2, 6 plus 4, minus 2, minus 8 and minus 6, minus 16 which is equal to the matrix containing elements 4, 10, minus 10 and minus 22. Now let us find the RHS which is AB into C. This is equal to the matrix containing elements 1, 3, 2, 4 into the matrix containing elements 2, 1, 0, minus 3 into the matrix containing elements 1, 0, minus 1, 1. The product of the matrices A and B will be a 2 by 2 matrix whose elements are the element in the first row first column is 1 into 2 plus 2 into 1. The element in the first row second column is 1 into 0 plus 2 into minus 3. The element in the second row first column is 3 into 2 plus 4 into 1 and the element in the second row second column is 3 into 0 plus 4 into minus 3 into the matrix containing elements 1, 0, minus 1, 1. This is equal to the matrix containing elements 2 plus 2, 6 plus 4, 0 minus 6 and 0 minus 12 into the matrix containing elements 1, 0, minus 1, 1. This is equal to the matrix containing elements 4, 10, minus 6, minus 12 into the matrix containing elements 1, 0, minus 1, 1. Now again the product of these two matrices is a matrix of order 2 by 2 whose elements are the element in the first row first column is 4 into 1 plus minus 6 into 0. The element in the first row second column is 4 into minus 1 plus minus 6 into 1. The element in the second row first column is 10 into 1 plus minus 12 into 0 and the element in the second row second column is 10 into minus 1 plus minus 12 into 1. This is equal to the matrix containing elements 4, 10, minus 10, minus 22. So, here we can see that the matrices obtained in the LHS and the RHS is the same. So, LHS is equal to RHS. This implies A into B, C is equal to AB into C. This completes our session. Hope you enjoyed the session.