Question

If \(A\) is a \(3 \times 3\) matrix such that \( |A| = 5 \), find the value of \( |adj(A)| \).

• A. \(5\)
• B. \(10\)
• C. \(25\)
• D. \(125\)

Hint:

For any \(n \times n\) matrix \(A\), \[ |adj(A)| = |A|^{n-1} \] This formula avoids the lengthy process of actually computing the adjoint matrix.

Answer 1

The Correct Option is C

Concept: For a square matrix \(A\) of order \(n\), an important determinant property involving the adjoint matrix is: \[ |adj(A)| = |A|^{\,n-1} \] where \( |A| \) is the determinant of matrix \(A\), and \(n\) is the order of the matrix. This property directly helps compute the determinant of the adjoint without explicitly finding the adjoint matrix.

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

Step 2: Apply the determinant property of the adjoint matrix. \[ |adj(A)| = |A|^{n-1} \] Substituting the given values: \[ |adj(A)| = 5^{3-1} \] \[ |adj(A)| = 5^2 \]

Step 3: Compute the value. \[ |adj(A)| = 25 \]