Let A = bmatrix x & 1 & 1 1 & x & 1 1 & 1 & x bmatrix , where x is an integer. If | adj (adj ( A 2 ) ) | = 16 , then the value of |x| is equal to

Step-by-step solution

Given A = bmatrix x & 1 & 1 1 & x & 1 1 & 1 & x bmatrix . First, let us find the determinant of A : |A| = x(x^2 - 1) - 1(x - 1) + 1(1 - x) |A| = x(x - 1)(x + 1) - (x - 1) - (x - 1) |A| = (x - 1)[x(x + 1) - 1 - 1] |A| = (x - 1)(x^2 + x - 2) |A| = (x - 1)^2(x + 2) We are given that | adj (adj ( A 2 ) ) | = 16 . For a 3 3 matrix M , |adj(adj(M))| = |M|^ (3-1)^2 = |M|^4 . Here, M = A 2 , so: | A 2 |^4 = 16 | A 2 | = 2 Using the scalar multiplication property for a 3 3 matrix, |kA| = k^3|A| . | A 2 | = ( 1 2 )^3 |A| = |