Question

If \(A\) is a square matrix of order \(3\) such that \( |adj\,A| = 64 \), then find the value of \( |A| \).

• A. \(4\)
• B. \(\pm 4\)
• C. \(8\)
• D. \(\pm 8\)

Hint:

For a square matrix of order \(n\), remember the identity \[ |adj\,A| = |A|^{\,n-1}. \] In many exam problems, once the order of the matrix is known, this formula allows you to directly relate the determinant of the matrix and its adjoint.

Answer 1

The Correct Option is D

Concept: For a square matrix \(A\) of order \(n\), the determinant of the adjoint matrix satisfies the property \[ |adj\,A| = |A|^{\,n-1} \] where \(n\) is the order of the matrix. This identity is frequently used in determinant problems involving adjoint matrices.

Step 1: Use the determinant property of the adjoint matrix. Since \(A\) is a square matrix of order \(3\), \[ |adj\,A| = |A|^{3-1} \] \[ |adj\,A| = |A|^{2} \]

Step 2: Substitute the given value. \[ |adj\,A| = 64 \] Therefore, \[ |A|^2 = 64 \]

Step 3: Solve for the determinant of \(A\). \[ |A| = \pm \sqrt{64} \] \[ |A| = \pm 8 \]