Question

If \( \det(AB) = \det(A)\det(B) \) and \( A \) is a non-singular matrix of order \( 3 \times 3 \), then \( \det(\text{adj}(A)) \) is:

• A. \( \det(A) \)
• B. \( (\det(A))^{-1} \)
• C. \( (\det(A))^2 \)
• D. \( (\det(A))^3 \)

Hint:

For any square matrix \( A \), the determinant of its adjugate matrix is given by \( \det(\text{adj}(A)) = (\det(A))^{n-1} \), where \( n \) is the order of the matrix.

Answer 1

The Correct Option is C

We are given that \( A \) is a non-singular matrix of order \( 3 \times 3 \), and we know the property: \[ \det(AB) = \det(A)\det(B) \] We are asked to find \( \det(\text{adj}(A)) \). For any square matrix \( A \), the determinant of its adjugate matrix is related to the determinant of \( A \) by the formula: \[ \det(\text{adj}(A)) = (\det(A))^{n-1} \] where \( n \) is the order of the matrix. Since \( A \) is a \( 3 \times 3 \) matrix, \( n = 3 \), so: \[ \det(\text{adj}(A)) = (\det(A))^{3-1} = (\det(A))^2 \] % Final Answer \[ \boxed{(\det(A))^2} \]