Let A and B be two matrices of order 3 3 such that A^2 B = 2I , where I is the identity matrix of order 3 . If (A) = 2 , then the value of (3 adj (2 adj (A B^2))) is equal to

Step-by-step solution

Given A^2 B = 2I , we take the determinant on both sides: (A^2 B) = (2I) ( (A))^2 (B) = 2^3 (I) Substituting (A) = 2 : 2^2 (B) = 8 4 (B) = 8 (B) = 2 Let M = A B^2 . (M) = (A) ( (B))^2 = 2 2^2 = 8 We need to evaluate (3 adj (2 adj (M))) . Using the properties (kX) = k^3 (X) and ( adj (X)) = ( (X))^2 for 3 3 matrices: ( adj (M)) = ( (M))^2 = 8^2 = 64 = 2^6 (2 adj (M)) = 2^3 ( adj (M)) = 8 2^6 = 2^9 ( adj (2 adj (M))) = ( (2 adj (M)))^2 = (2^9)^2 = 2¹⁸ Finally, applying the scalar multiple property for the outermost f