 Hi, this is Meenu and going to help you to solve this question. It says for the matrix A, find the numbers A and B such that A squared plus A A plus B I is equal to 0 and hence find A inverse. So before moving on to the solution, let us first understand some key ideas. The key idea is if we multiply a matrix A with its inverse, it gives us the identity matrix. And if we multiply any matrix A with the identity matrix, it gives us the matrix A itself. Keeping these key ideas in mind, let's move on to the solution. The matrix A is 3, 2, 1, 1. We need to find A squared A A plus B I. So A squared is A into A which is equal to 3 into 3 plus 2 into 1 3 into 3 plus 2 into 1 3 into 2 plus 2 into 1 1 into 3 plus 1 into 1 1 into 2 plus 1 into 1 and it is equal to 9 plus 2 is 11 6 plus 2 is 8 3 plus 1 is 4 2 plus 1 is 3. We now find A A A into the matrix A 3, 2, 1, 1 which is equal to 3A plus 2A 3A to A We just have to multiply A with each term. We now find B I. I is the identity matrix of order 2 cross 2 is 1, 0, 0, 1. So it is equal to B 0, 0, B Now We are given that A squared plus A A plus B I is 0. A squared plus A A plus B I is equal to 0. That implies A squared 11 8, 4, 3 plus A A which is 3A 2A A A plus B 0, 0, B is equal to 0 matrix and again this is equal to adding the 3 we will get 11 plus 3A plus B 8 plus 2A We just add the corresponding terms of each matrix plus 0 4 plus A plus 0 3 plus A plus B is equal to the 0 matrix. Now we know that two matrices are equal if their corresponding terms are equal. So we equate any two of the following equation and we'll solve them for A and B. So from this we'll get A plus 4 is equal to 0 that implies A equal to minus 4 and 3A plus A plus B is equal to 0 implies 3 minus 4 plus B is equal to 0 and that implies B is equal to 1. So hence A is equal to minus 4 and B is equal to minus 1. Now we find the inverse. Now we are given that A square plus A A, A is minus 4 that is minus 4A B is 1. So we have 1 here. B is 1. So it is 1 I is equal to 0 and which can be written as A into A minus 4A plus I is equals to 0 and which again implies A into A minus 4A is equal to minus I. Now we have to find the value of A inverse. So we post-multiply this equation by A inverse post multiply 1 by A inverse. So we'll get A into A into A inverse minus 4 into A A inverse is equals to minus I into A inverse. And which gives us A into A into A inverse is I minus 4 I is equals to minus A inverse because we know that any matrix multiplied with identity matrix gives us the matrix itself. And hence this reduces to A minus 4I is equals to minus A inverse and that again implies A inverse is equal to 4I minus A. Now we can solve this easily. That implies A inverse is equals to 4I. That implies 4, 0, 0, 4. 4I is 4, 0, 0, 4 minus A which is 3, 2, 1, 1. So it becomes equal to 4 minus 3 is 1 minus 2 minus 1. 4 minus 1 is 3. Hence the value of A is minus 4, B is 1 and A inverse is equal to 1 minus 2 minus 1 3. So this completes the question. Bye for now. Take care. Hope you enjoyed. See you in the next study session.