 Hi and welcome to our session. Let us discuss the following question. The question says if A is equal to this matrix, prove that A squared minus 4A minus 5I is equal to 0, hence find A inverse. Let us now begin with the solution. We are given that A is equal to 3 by 3 matrix in which elements are 1, 2, 2, 2, 1, 2, 2, 2, 1. We have to first prove that A squared minus 4A minus 5I is equal to 0 and then we have to find A inverse. Let us first find A squared. Now A squared is equal to A into A, that is this matrix multiplied by itself. This is equal to 3 by 3 matrix in which elements are 1 into 1 plus 2 into 2 plus 2 into 2, 1 into 2 plus 2 into 1 plus 2 into 2, 1 into 2 plus 2 into 2 plus 2 into 1. Now in second row we will have 2 into 1 plus 1 into 2 plus 2 into 2, 2 into 2 plus 1 into 1 plus 2 into 2, 2 into 2 plus 1 into 2 plus 2 into 1, and in third row we will have 2 into 1 plus 2 into 2 plus 1 into 2, 2 into 2 plus 2 into 1 plus 1 into 2, 2 into 2 plus 2 into 1 plus 2 into 2 plus 2 into 2 plus 1 into 1. This is equal to 3 by 3 matrix. That is A, to 3 by 3 matrix in which elements are 9, 8, 8 square minus 4A minus 5 at 0, 0, 5, 0, 0, 0. This is equal to 3 by 3 matrix in which elements are 4, minus 5, 8, minus 8, minus 0, minus 0, minus 0, 9, minus 4, minus 4, sorry, all elements are 0, into 0. Inverse I is equal to 0. This implies A inverse A into A minus 5 A inverse into I is A inverse equals to 0. Again A inverse A is minus 5 A inverse equals to 0. This implies A minus 4I minus 5A inverse is equal to 0. This implies minus 4I is equal to 3 by 3 matrix in which elements are is equal to this matrix. Now we will multiply 0, 0, 0, 4, 0, 0, 0, 4. This is equal to 3 by 3 matrix in which elements are 3. So 5A inverse is equal to this matrix and this implies is equal to 1 by 3.