 Hello and welcome to the session. In this session we discussed the following question which says if a is equal to matrix of order 3 by 3 with elements 1, 2, 2, 2, 1, 2, 2, 2, 1, verify that a square minus 4a minus 5i is equal to 0. Let's move on to the solution. We are given matrix a is equal to 1, 2, 2, 2, 1, 2, 2, 2, 1. Now we have a square is equal to a into a. Now a matrix is this. So we have a square is equal to the matrix 1, 2, 2, 2, 1, 2, 2, 2, 2, 1 into the matrix a itself that is 1, 2, 2, 2, 1, 2, 2, 2, 1. Now we need to multiply both these matrices. So we multiply these elements with these elements. This would be equal to Now the entry in the first row first column would be 1 into 1 plus 2 into 2 plus 2 into 2. This would be equal to 9. Then to find the entry in the first row second column we multiply this row of this matrix with this column of the second matrix. So this gives us 1 into 2 plus 2 into 1 plus 2 into 2 which would be equal to 8. Now we find the entry in the first row third column. We would get this by multiplying this row of the first matrix with this column of the second matrix. That is we have 1 into 2 plus 2 into 2 plus 2 into 1 which would be equal to 8. Now to find the entry in the second row first column we multiply the second row of this matrix with the first column of this matrix. So we get 2 into 1 plus 1 into 2 plus 2 into 2 which gives us 8. Then to find the entry in the second row second column we multiply the second row of this matrix with the second column of the second matrix. So we get 2 into 2 plus 1 into 1 plus 2 into 2 which gives us 9. Now to find the entry in the second row third column we multiply the second row of this matrix with the third column of this matrix. So we get 2 into 2 plus 1 into 2 plus 2 into 1 this gives us 8. Next we need to find the entry in the third row first column for this we multiply this third row of the first matrix with the first column of the second matrix. So we get 2 into 1 plus 2 into 2 plus 1 into 2 this gives us 8. Now we have to find the entry in the third row second column for this we multiply the third row with the second column of this matrix. So we get 2 into 2 plus 2 into 1 plus 1 into 2 which gives us 8. Now to find the entry in the third row third column we multiply the third row of this matrix by the third column of this matrix. So we get 2 into 2 plus 2 into 2 plus 1 into 1 which gives us 9. So we get a square is equal to the matrix of order 3 by 3 with elements 9, 8, 8, 8, 9, 8, 8, 8, 9. Now considering this equation you can see that we have found out a square and next step would be to find out minus 4 a. So we have minus 4 a is equal to minus 4 into the matrix a that is 1, 2, 2, 2, 1, 2, 2, 2, 1. Now we know that when we multiply a matrix with a scalar then each element of the given matrix is multiplied by that scalar. So this would give us minus 4 into 1 that is minus 4, minus 4 into 2 that is minus 8, minus 4 into 2, minus 8, minus 4 into 2, minus 8, minus 4 into 1 give us minus 4, minus 4 into 2 again is minus 8. Then minus 4 into 2, minus 8, minus 4 into 2 is minus 8, minus 4 into 1 is minus 4. So this is the matrix minus 4 a. Then next we have minus 5 a so this would be equal to minus 5 into the identity matrix that is 1, 0, 0, 0, 1, 0, 0, 0, 1. So this would be equal to minus 5 into 1 that is minus 5 minus 5 into 0 is 0 minus 5 into 0 is again 0, minus 5 into 0 is 0, minus 5 into 1 is minus 5, minus 5 into 0 is 0, minus 5 into 0 is again 0. Then we have minus 5 into 0 which gives us 0 minus 5 into 1 is minus 5. So this is minus 5 I. Now let's consider the given equation that is a square minus 4a minus 5y. So this would be equal to the matrix that is 9, 8, 8, 8, 9, 8, 8, 8, 8, 9. So we are substituting the value for a square in this equation then minus 4a so this would be plus then the value for minus 4a which is minus 4 minus 8 minus 8 minus 8 minus 4 minus 8 then minus 8 minus 8 minus 4 then plus the value for minus 5y which is minus 5 0 0 0 minus 5. Now we need to add these three matrices so this would be equal to now the first element of the first row would be 9 minus 4 minus 5. Second element of the first row is 8 minus 8 plus 0 then the third element of the first row is 8 minus 8 plus 0 then the third element of the first element of the second row is 8 minus 8 plus 0 second element of the second row is 9 minus 4 minus 5 then the third element of the second row is 8 minus 8 plus 0 first element of the third row is 8 minus 8 plus 0 then the second element of the third row is 8 minus 8 plus 0. Third element of the third row is 9 minus 4 minus 5. So this is equal to 0, 0, 0, 0, 0, 0, 0, 0. That is equal to 0. That is we get a square minus 4a minus 5i is equal to 0. Hence we have a square minus 4a minus 5i equal to 0. So this completes the session. Hope you have understood the solution for this question.