Question

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

• A. \(5\)
• B. \(25\)
• C. \(125\)
• D. \(15\)

Hint:

For an \(n \times n\) matrix, always remember the identity \( |adj(A)| = |A|^{n-1} \). For example, if the matrix is \(3 \times 3\), then \( |adj(A)| = |A|^2 \).

Answer 1

The Correct Option is B

Concept: For any square matrix \(A\) of order \(n\), an important determinant property related to the adjoint matrix is: \[ |adj(A)| = |A|^{\,n-1} \] where \( |A| \) = determinant of matrix \(A\), \( adj(A) \) = adjoint of matrix \(A\), \( n \) = order of the square matrix. This identity is frequently used in determinant and inverse-related problems.

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

Step 2: Use the determinant property of adjoint. \[ |adj(A)| = |A|^{n-1} \] Substitute the given values \( |A| = 5 \) and \( n = 3 \): \[ |adj(A)| = 5^{3-1} \] \[ |adj(A)| = 5^2 \] \[ |adj(A)| = 25 \]