 Hello and welcome to the session. In this session we are going to discuss the following question and the question says that if matrix x is equal to matrix containing elements 3-1, 1, 2 show that 5x-x square is equal to 7i where i is the identity matrix. We are given that the matrix x is equal to matrix containing elements 3-1, 1, 2 and we need to show that 5x-x square is equal to 7i. For this first we shall find x square which will be equal to x into x that is equal to the matrix containing elements 3-1, 1, 2 into the same matrix containing elements 3-1, 1, 2. On multiplying these two matrices we will obtain a 2 by 2 matrix in which the first row first column element will be obtained by multiplying the first row of the first matrix by the first column of the second matrix that is 3 into 3 plus 1 into minus 1. The element in the first row second column is obtained by multiplying the first row of the first matrix by the second column of the second matrix that is 3 into 1 plus 1 into 2. The element in the second row first column is obtained by multiplying the second row of the first matrix by the first column of the second matrix that is minus 1 into 3 plus 2 into minus 1. And lastly the element in the second row second column is obtained by multiplying the second row of the first matrix by the second column of the second matrix that is minus 1 into 1 plus 2 into 2. This will be equal to the matrix containing elements 9 minus 1 minus 3 minus 2, 3 plus 2 and minus 1 plus 4. This will be equal to matrix containing elements 8 minus 5, 5 and 3. So now we have x square is equal to matrix containing elements 8 minus 5, 5, 3. We shall now find 5x. 5x will be equal to 5 into the matrix containing elements 3 minus 1, 1, 2. If we multiply a scalar by a matrix it has to be multiplied by every element of the matrix. So this is equal to the matrix containing elements 5 into 3, 5 into 1, 5 into minus 1 and 5 into 2. That is equal to the matrix containing elements 15 minus 5, 5 and 10. We had to show that 5x minus x square is equal to 7i. So consider the LHS which is 5x minus x square. This will be equal to the matrix containing elements 15 minus 5, 5, 10 minus the matrix containing elements 8 minus 5, 5, 3. To subtract 2 matrices we have to subtract the corresponding elements of the matrices. So this will be equal to the matrix containing elements 15 minus 8, 5 minus 5, minus 5, minus 5 and 10 minus 3. Which is equal to the matrix containing elements 7, 0, 0 and 7. As we can see 7 is common so we take 7 out and we get 7 into the matrix containing elements 1, 0, 0, 1. This is equal to 7i where i is equal to the matrix containing elements 1, 0, 0, 1 which is the identity matrix. So here we can see that 5x minus x square is equal to 7i. This completes our session. Hope you enjoyed the session.