Let A and B be two 3 3 matrices with real entries such that A^2 = 3B and |A| = 9 . If X = 2AB⁻¹ and |adj(adj(X))| = 2^x 3^y , then the value of x + y is equal to

Step-by-step solution

Given A and B are 3 3 matrices, A^2 = 3B , and |A| = 9 . Taking the determinant on both sides of A^2 = 3B : |A^2| = |3B| |A|^2 = 3^3 |B| (since B is a 3 3 matrix) 9^2 = 27 |B| 81 = 27 |B| |B| = 3 Now, we are given X = 2AB⁻¹ . Taking the determinant on both sides: |X| = |2AB⁻¹| = 2^3 |A| |B⁻¹| |X| = 8 |A| 1 |B| |X| = 8 9 1 3 = 24 Prime factorising 24 , we get |X| = 2^3 3^1 . We know that for an n n matrix M , |adj(adj(M))| = |M|^ (n-1)^2 . Here, X is a 3 3 matrix, so n = 3 . |adj(adj(X))| = |X|^ (3-1)^2 = |X|^4 |adj