 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 the 2 by 2 matrix minus 2, 1, 3, minus 3, find A cube. We have to find A cube. A cube can be written as A square into A. So to find A cube we have to first find A square. Therefore A square is equal to A into A which will be equal to minus 2, 1, 3, minus 3 into the matrix minus 2, 1, 3, minus 3. To multiply two matrices the first step is to find the first row and the first column. Now to find the first row and the first column we multiply the first row of the first matrix with the first column of the second matrix that is minus 2 into minus 2 plus 3 into 1. The rest of the elements we will find in the further steps. Therefore the first element is 4 plus 3 which is equal to 7. Now we have already got the element in the first row first column that is 7. Now we have to find the next element that is the element in the first row second column and this is obtained by multiplying the first row of the first matrix by the second column of the second matrix. So the element in the first row second column will be minus 2 into 3 plus 3 into minus 3 which is equal to minus 6 plus minus 9 which will be equal to 7 and minus 15. So from the first and the second step we have obtained two elements that is the element in the first row first column and the element in the first row second column. Now we have already got two elements in the matrix 7 and minus 15 and in the third step we find out the element in the second row and first column. So for this we multiply the elements in the second row of the first matrix with the first column of the second matrix that is 1 into minus 2 plus minus 3 into 1. This is equal to 7 minus 15 this becomes minus 2 minus 3 which will be equal to minus 5. So now we have the three elements 7 minus 5 minus 15 and we need to find the second row second column element. This is obtained by multiplying the second row of the first matrix by the second column of the second matrix that is 1 into 3 plus minus 3 into minus 3 which is equal to 3 plus 9 which is equal to the matrix 7 minus 5 minus 15 and 12. So from all four steps we obtain a square is equal to the 2 by 2 matrix 7 minus 5 minus 15 12. So now as a cube is equal to a square into a therefore a cube is equal to the matrix 7 minus 5 minus 15 12 into minus 2 1 3 minus 3. Again following the same steps we have to find the product of matrices a square and a. So 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. So we have 7 into minus 2 plus minus 15 into 1. Then the element of the first row second column is obtained by multiplying the elements in the first row of the first matrix and the second column of the second matrix. So we have 7 into 3 plus minus 15 into minus 3. Now we have to obtain the element in the second row first column which is obtained by multiplying the second row of the first matrix by the first column of the second matrix that is minus 5 into minus 2 plus 12 into 1. Now lastly we have to find the element in the second row second column which is obtained by multiplying the second row of the first matrix by the second column of the second matrix that is minus 5 into 3 plus 12 into minus 3. So here the first element is minus 14 minus 15. The second element is 21 plus 45. The third element is 10 plus 12 and the fourth element is minus 15 minus 36. So this is equal to minus 29, 22, 66 and minus 51. So we obtain a cube is the 2 by 2 square matrix minus 29, 22, 66 and minus 51 which is our answer. This completes our session. Hope you enjoyed the session.