Question

If \( A \) is a square matrix of order \( 3 \times 3 \), \( \det A = 3 \), then the value of \( \det(3A^{-1}) \) is:

• A. 3
• B. 27
• C. 9
• D. \( \frac{1}{3} \)

Hint:

For a matrix \( A \) of order \( n \), \( \det(kA) = k^n \det(A) \). If \( A^{-1} \) is the inverse of \( A \), \( \det(A^{-1}) = \frac{1}{\det(A)} \).

Answer 1

The Correct Option is C

The property of determinants states that for a square matrix \( A \), the determinant of \( kA \) is \( k^n \det A \), where \( n \) is the order of the matrix. \[ \det(3A^{-1}) = 3^3 \det(A^{-1}) = 27 \times \frac{1}{\det A} = 27 \times \frac{1}{3} = 9 \] Thus, the value of \( \det(3A^{-1}) \) is \( 9 \).