Question

If \( B = \begin{bmatrix} 1 & \alpha & 3 \\ 1 & 3 & 3 \\ 2 & 4 & 4 \end{bmatrix} \) is the adjoint of a \( 3 \times 3 \) matrix \( A \) and \( |A| = 4 \), then the value of \( \alpha \) is:

• A. 4
• B. 7
• C. 9
• D. 11
• E. 13

Hint:

The determinant of the adjoint of a matrix is equal to the square of the determinant of the original matrix.

Answer 1

The Correct Option is D

The determinant and adjoint relation is: \[ {det}({adj}(A)) = |A|^2 \] Since \( |A| = 4 \), we compute: \[ {det}(B) = 4^2 = 16 \] Expanding the determinant of \( B \), solving for \( \alpha \), and setting it equal to 16 gives: \[ \alpha = 11 \]