Question

If \(A\) is a \(3\times3\) matrix such that \(|A| = 4\) and \(B = \text{adj}\,A\), find the value of \(|B|\).

• A. \(4\)
• B. \(8\)
• C. \(16\)
• D. \(64\)

Hint:

For an \(n \times n\) matrix: \[ |\text{adj}(A)| = |A|^{\,n-1} \] Special cases:
• If \(A\) is \(2\times2\): \(|\text{adj}(A)| = |A|\)
• If \(A\) is \(3\times3\): \(|\text{adj}(A)| = |A|^2\) This property is frequently used in determinant and inverse matrix problems.

Answer 1

The Correct Option is C

Concept: For a square matrix \(A\) of order \(n\), an important property of determinants and adjoint matrices is: \[ |\,\text{adj}(A)\,| = |A|^{\,n-1} \] where:
• \(A\) is an \(n \times n\) matrix
• \(\text{adj}(A)\) is the adjoint (adjugate) of matrix \(A\)
• \(|A|\) represents the determinant of matrix \(A\) Thus, the determinant of the adjoint matrix depends on the determinant of the original matrix and the order of the matrix.

Step 1: Identify the order of the matrix. The matrix \(A\) is given as a \(3 \times 3\) matrix. Therefore, \[ n = 3 \]

Step 2: Use the determinant property of adjoint matrices. Using the formula: \[ |\text{adj}(A)| = |A|^{\,n-1} \] Substitute \( |A| = 4 \) and \( n = 3 \): \[ |\text{adj}(A)| = 4^{3-1} \] \[ |\text{adj}(A)| = 4^2 \]

Step 3: Compute the value. \[ 4^2 = 16 \] Thus, \[ |B| = |\text{adj}(A)| = 16 \] \[ \boxed{16} \]