Question

If |A| = 2, B = adj(adj(2A)), A is a 3×3 matrix, and tr(A) = 3, what is tr(B) + |B|?

• A. 4
• B. 6
• C. 2
• D. 8

Hint:

The determinant and trace of a matrix can be used to simplify problems involving adjugates and scalar multiples of matrices.

Answer 1

The Correct Option is B

To solve this, recall the properties of determinants and adjugates. If \( A \) is a 3×3 matrix, then \( |\text{adj}(A)| = |A|^2 \). 

Similarly, for the adjugate of a scalar multiple of a matrix, use the identity \( \text{adj}(kA) = k^{n-1} \text{adj}(A) \), 
where \( n \) is the order of the matrix. 

Solve for \( B \) and then compute the trace and determinant to find the solution.