Let A be a 3 3 matrix such that |A|=2 . If B = adj (2 adj (A)) and |B| + trace (B) = 1064 , then the value of trace (A) is :

Step-by-step solution

For any n n matrix X and scalar k , we know that adj (kX) = k^ n-1 adj (X) . Since A is a 3 3 matrix, adj (A) is also a 3 3 matrix. Therefore, taking k=2 and X= adj (A) , we have: B = adj (2 adj (A)) = 2³⁻¹ adj ( adj (A)) = 4 adj ( adj (A)) Using the property adj ( adj (A)) = |A|^ n-2 A , for n=3 we get: adj ( adj (A)) = |A|A Substitute |A| = 2 into the expression for B : B = 4(|A|A) = 4(2A) = 8A Now, calculate the determinant and trace of B : |B| = |8A| = 8^3|A| = 512 2 = 1024 trace (B) = trace (8A) = 8 trace (A)