 Hi, and welcome to the session. I am Shashi and I am going to help you to solve the following question. Question is for the matrices A and B, verify that transpose of AB is equal to B transpose A transpose, where A is equal to matrix 0, 1, 2 and B is equal to matrix 157. First of all, let us understand the key idea to solve the given question. If A is equal to Aij Bn m into n matrix, then the matrix obtained by interchanging the rows and columns of A is called transpose of A. The transpose of the matrix A is denoted by A dash. In other words, if A is equal to matrix Aij of the order m into n, then A transpose or A dash is equal to matrix Aji of order n into m. Let us now start with the solution. We are given A is equal to matrix 0, 1, 2 and B is equal to matrix 157. We can see A is of the order 3 into 1 since it is having 3 rows and 1 column and B is of the order 1 into 3 as it is having 1 row and 3 columns. Now we can see the number of columns of A is equal to number of rows of B that is equal to 1. So, their multiplication is defined. So, A B is equal to matrix 0, 1, 2 multiplied by 157. Now, using multiplication of matrices we get A B is equal to matrix. First element of first row is 0 multiplied by 1. Second element of first row is 0 multiplied by 5. Third element of first row is 0 multiplied by 7. Similarly, first element of second row is 1 multiplied by 1. Second element of second row is 1 multiplied by 5. Third element of second row is 1 multiplied by 7. Similarly, first element of third row is 2 into 1. Second element of third row is 2 into 5. And third element of the third row is 2 into 7. So, we can write A v is equal to matrix 0, 0, 0, 1, 5, 7, 2, 10, 14, now we will find transpose of A v transpose of A v is equal to 0, 0, 0, 1, 5, 7, 2, 10, 14, we have obtained the transpose of A v by interchanging the rows and columns of A v. Now A is equal to matrix 0, 1, 2 and v is equal to matrix 1, 5, 7 it is already given in the question. So, we can write A transpose is equal to matrix 0, 1, 2, we can obtain A transpose by interchanging the rows and columns of A. Then similarly we can obtain B transpose, B transpose would be equal to 1, 5, 7, we can obtain B transpose by interchanging the rows and columns of B. Now we know A transpose is having the order 1 into 3 as it is having 1 row and 3 columns and B transpose is of the order 3 into 1 as it is having 3 rows and 1 column. Now we have to calculate B transpose A transpose. So, the number of columns of B transpose must be equal to the number of rows of A transpose only then their multiplication would be defined. Now we can see the number of columns in B transpose is 1 and the number of rows in A transpose is also equal to 1. So, their multiplication is defined. So, let us now find out. B transpose A transpose that is matrix 1, 5, 7 multiplied by matrix 0, 1, 2. So, B transpose A transpose is equal to matrix 1 multiplied by 0, 1 multiplied by 1, 1 multiplied by 2. Then 5 multiplied by 0, 5 multiplied by 1, 5 multiplied by 2, 7 multiplied by 0, 7 multiplied by 1, 7 multiplied by 2. So, B transpose A transpose is equal to matrix 0, 1, 2, 0, 5, 10, 0, 7, 14. But this is equal to A B transpose as we can see here. So, we can write A transpose is equal to transpose of A B. Therefore, our required answer is transpose of A B is equal to B transpose A transpose. Hence, verify it. This completes the session. Hope you enjoyed the session. Take care and good bye.