 Hi, this is Meenu and going to help you to solve this question it says if A is this 2 cross 2 matrix show that A square minus 5A plus 7I is equal to 0 hence find A inverse. So let us first understand the key idea to solve this problem or you can say a note that A into A inverse that is product of a matrix with its inverse gives us the identity matrix. So keeping this note in your mind we will solve this question. We first show that A square minus 5A plus 7I is 0. So A is 3 1 minus 1 2 then A square is given by A into A which is equal to 3 into 3 plus 1 into minus 1 3 into 1 plus 1 into 2 minus 1 into 3 plus 2 into minus 1 minus 1 into 1 plus 2 into 2 and it is equal to 9 minus 1 8 3 plus 2 is 5 minus 3 minus 2 is minus 5 4 minus 1 is 3. Now we have to find 5A so 5A is equal to 15 5 minus 5 and 10 then we need to find 7I where I is the identity matrix of order 2 cross 2 so it is 7 0 0 7. Now we have to show that A square minus 5A plus 7I is 0 0 means the 0 matrix so A square minus 5A plus 7I is equal to minus 15 5 minus 5 10 plus 7 0 0 7 and it is equal to 8 minus 15 plus 7 5 minus 5 5 minus 5 plus 0 minus 5 minus minus plus 5 plus 0 3 minus 10 plus 7 and we can see that it is the 0 matrix so hence it is proved that A square minus 5A plus 7I is 0. Now we have to find the inverse. Now we know that A square minus 5A plus 7I is 0 and it can also be written as A into A minus 5A is equal to minus 7I. Now we need to find the inverse of A so we post multiply by A inverse so we will get A into A inverse minus 5A into A inverse minus is equal to minus 7I into A inverse. Now we know that A into A inverse is identity matrix that is I minus 5I is equal to minus 7 into A inverse because any matrix multiplied with identity matrix is the matrix itself so we get A into I is A minus 5I is equal to minus 7A inverse that implies A inverse is equal to minus 1 upon 7 into A minus 5I 5I dividing both sides by minus 7 is that right? Now we can easily solve this minus 1 upon 7 into A is 3 1 minus 1 2 minus 5I 5I is 5 0 0 5 and again this is equal to minus 1 upon 7 minus 2 1 minus 1 minus 3 and which is also equal to 1 upon 7 2 minus 1 1 3 multiplying minus 1 with all the entries we will get this. Hence A inverse is equal to 1 upon 7 into 2 1 minus 1 3 which is the answer so this completes the question. Bye for now take care hope you enjoyed the session.