 Hi, this is Meenu and going to help you to solve this question. It says if a is this 3 x 3 matrix, verify that a cube minus 6a square plus 9a minus 4i is 0 and hence find a inverse. So let's move on to the solution. The given matrix A is, now to prove a cube minus 6a square plus 9a minus 4i is 0 we need to find a cube 6a square and 9a. So we find a square which is a into a into a again which is 2 minus 1 1 minus 1 2 minus 1 1 minus 1 2 and multiplying the two matrices 2 into 2 plus minus 1 into minus 1 is 1 plus 1 into 1 minus 2 minus 1 2. Minus 2 minus 1 2 minus 1 into minus 1 is plus 1 plus 1 plus 2 minus 2 minus 2 minus 1 1 plus 4 2 into 2 is 4 plus 1 minus 1 minus 4. 1 minus 2 minus 2 then 2 plus 1 plus 2 then minus 1 minus 2 minus 2 then 1 into 1 is 1 minus 1 into minus 1 is plus 1 2 into 2 is 4 and hence a square is equal to 6 minus 5 5 minus 5 6 minus 5 5 minus 5 6. We now find a cube which is equal to a square into a which is a square is 6 minus 5 5 minus 5 6 minus 5 5 minus 5 6 into a 2 minus 1 1 minus 1 2 minus 1 1 minus 1 2 and multiplying the two matrices in the similar manner will get the product as 22 minus 21 21 minus 21 22 minus 21 21 minus 21 22. Now we have to find a cube minus 6 a square plus 9 a minus 4 i. Now a cube is 22 minus 21 21 minus 21 22 minus 21 21 minus 21 22 minus 6 a square that is multiplying each term of a square with 6 gives us 6 into 6 is this is a square 6 into 6 is 36 6 into minus 5 is minus 30 6 into 5 is 30 then minus 30 36 minus 30 30 minus 30 36 plus 9 a multiplying each term of a with 9 a is this so we multiply each term of a with 9 gives us 18 minus 9 9 minus 9 18 minus 9 9 minus 9 18 minus 4 i where i is the identity matrix of order 3 cross 3 so it is 4 0 0 0 4 0 0 0 4 and which is again equal to 22 minus 36 plus 18 minus 4 minus 21 minus minus plus 30 minus 21 plus 30 minus 9 plus 0 we just add and subtract the corresponding terms of the matrices 21 minus 30 plus 9 then minus 21 plus 30 minus 9 minus 0 then 22 minus 36 minus 36 plus 18 minus 4 then minus 21 minus minus plus 30 plus minus minus 9 minus 0 then 21 minus 30 plus 9 minus 0 then minus 21 plus 30 minus minus is plus plus 30 then minus then minus 9 minus 0 then 22 minus 36 plus 18 minus 4 and we can see that this matrix is equal to a 0 matrix this is what we had to prove that a cube minus 6 a square plus 9 a minus 4 i is equal to 0 now we have to find the inverse of the matrix using equation a cube minus 6 a square plus 9 a minus 4 i is equal to 0 now it can be written as a square into a minus 6 into a into a plus 9 a is equal to 4 i now we have to find the inverse so we'll post multiply by a inverse so we'll get a square into a into a inverse minus 6 a into a into a inverse plus 9 into a into a inverse is equal to 4 i into a inverse 4 i into a inverse and we know that a into a inverse multiplying any matrix with its inverse gives us identity matrix and multiplying any matrix with the identity matrix gives us the matrix itself so using this formula we'll get a square a into a inverse is i minus 6 a into i plus 9 i is equal to i into a inverse is a inverse so it is 4 a inverse and again it is equal to a square because a square into i is a square minus 6 a plus 9 i is equal to 4 a inverse and it again implies a inverse is equal to 1 upon 4 into a square minus 6 a plus 9 i now we'll find the inverse of a using this equation so a inverse is equal to 1 by 4 into a square and a square is 6 minus 5 5 minus 5 6 minus 5 5 minus 5 6 minus 6 a and 6 a is obtained by multiplying each element of a with 6 so that it gives us 12 minus 6 6 minus 6 12 6 6 minus 6 12 plus 9 i 9 i means multiplying identity matrix with 9 so that it gives us 9 0 0 0 9 0 0 0 9 we now solve this so it becomes 1 by 4 into 6 minus 12 plus 9 minus 5 minus minus plus 6 minus 5 plus 6 plus 0 five minus six minus six plus zero minus five plus six plus zero six minus 12 plus 9 then minus five plus six plus zero then five minus 6 plus 0 then minus 5 minus minus plus 6 plus 0 then 6 minus 12 plus 9 and hence it is equal to 1 by 4 into 3 1 minus 1 1 3 1 minus 1 1 3 so this is the A inverse hence A inverse is equal to 1 by 4 3 1 minus 1 1 3 1 minus 1 1 3 which is the answer so this completes the question. Bye for now, take care, hope you enjoyed this session.