 Hi and welcome to the session I am Shashi and I am going to introduce all the following question. Question is if A is equal to matrix minus 1, 2, 3, 5, 7, 9, minus 2, 1, 1 and B is equal to matrix minus 4, 1, minus 5, 1, 2, 0, 1, 3, 1 then refer that A minus B whole transpose is equal to A transpose minus B transpose. First of all let us understand the key idea to solve the given question. If A is equal to matrix A ij B n 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 transpose that is A dash in other words if A is equal to matrix A ij of the order m into n then A transpose that is A dash is equal to matrix A ji of the order n into m. Let us now start with the solution. We have given A is equal to matrix minus 1, 2, 3, 5, 7, 9, minus 2, 1, 1 B is equal to matrix minus 4, 1, minus 5, 1, 2, 0, 1, 3, 1 that is minus B. We know minus B is equal to we will multiply every element of matrix B with minus 1 to get minus B. So, we get minus B is equal to matrix 4, minus 1, 5, minus 1, minus 2, 0, minus 1, minus 3, minus 1. Now we know A minus B is equal to A plus minus B. So, we will get A plus minus B equal to matrix minus 1, 2, 3, 5, 7, 9, minus 2, 1, 1 plus matrix minus A that is 4, minus 1, 5, minus 1, minus 2, 0, minus 1, minus 3, minus 1. So, we get A minus B is equal to minus 1 plus 4, 2 plus 1, 3 plus 5, 5 plus minus 1, 7 plus minus 2, 9 plus 0, minus 2, plus minus 1, 1 plus minus 1. Here we had added the corresponding elements of the two matrices. So, we will get 1, 8, 5, 9, minus 3, minus 2, 0 as the matrix equal to A minus E. A minus B whole transpose is equal to 3, 1, 8, 4, 5, 9, minus 3, minus 2, 0. First row of A minus B will change into first column of A minus B whole transpose. Second row of A minus B will change into second column of A minus B whole transpose and third row of A minus B will change into third column of A minus B whole transpose. So, A minus B whole transpose is equal to matrix 3, 4, minus 3, 1, 5, minus 2, 8, 9, 0. Now, A given to us in the question is matrix minus 1, 2, 3, 5, 7, 9, minus 2, 1, 1. So, A transpose is equal to matrix minus 1, 2, 3, 5, 7, 9, minus 2, 1, 1. Here first row of A will change into first column of A transpose, second row of A will change into second column of A transpose, third row of A will change into third column of A transpose. So, A transpose is equal to matrix minus 1, 5, minus 2, 2, 7, 1, 3, 9, 1. Now, B is equal to matrix minus 4, 1, minus 5, 1, 2, 0, 1, 3, 1. So, B transpose is equal to matrix minus 4, 1, minus 5, 1, 2, 0, 1, 3, 1. First row of B has changed into first column of B transpose, second row of B has changed into second column of B transpose and third row of B has changed into third column of B transpose. So, B transpose is equal to minus 4, 1, 1, 1, 2, 3, minus 5, 0, 1, matrix. Now, we will find minus B transpose minus B transpose is equal to matrix 4, minus 1, minus 1, minus 1, minus 2, minus 3, 5, 0, minus 1. To obtain minus B transpose we have multiplied every element of B transpose with minus 1. Now, we have to find A transpose minus B transpose which is equal to A plus A transpose plus minus B transpose, right. So, we can write A transpose minus B transpose is equal to matrix minus 1, 5, minus 2, 2, 7, 1, 3, 9, plus minus B transpose, minus B transpose is 4, minus 1, minus 1, minus 1, minus 2, minus 3, 5, 0, minus 1. Now, let us add the two matrices. To add the matrices we will add the corresponding elements of the two matrices. So, we get minus 1 plus 4, 5 plus minus 1, minus 2, plus minus 1, 2 plus minus 1, 7 plus minus 3, 1 plus minus 3, 3 plus 5, 9 plus 0, 1 plus minus 1. We can write A transpose minus B transpose is equal to matrix 3, 4, minus 3, 1, 5, minus 2, 8, 9, 0. So, A transpose minus B transpose is equal to matrix 3, 4, minus 3, 1, 5, minus 2, 8, 9, 0. But this is equal to A minus B whole transpose as we can see here. Let us name this as 1, this as 2. From expressions 1 and 2, we get A minus B whole transpose is equal to A transpose minus B transpose, verified. Hence, we have verified the given expression in the question. So, this completes the session. Hope you understood the session. Take care and goodbye.