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

Step 1: Understanding the Question:
We are given the determinant of the adjoint of a matrix \(A\) and its order \(n\).
We need to calculate the determinant of the original matrix \(A\).
Step 2: Key Formula or Approach:
For any square matrix \(A\) of order \(n\), the determinant of its adjoint is related to the determinant of the matrix by the identity:
\[ |adj\,A| = |A|^{\,n-1} \] Step 3: Detailed Solution:
We are given that the order of the matrix is \(n = 3\).
Substituting \(n = 3\) into the property formula, we get:
\[ |adj\,A| = |A|^{3-1} \] \[ |adj\,A| = |A|^{2} \] We are also given that \(|adj\,A| = 64\).
Substituting this value into our equation yields:
\[ |A|^2 = 64 \] Taking the square root on both sides to solve for \(|A|\):
\[ |A| = \pm \sqrt{64} \] \[ |A| = \pm 8 \] Step 4: Final Answer:
The determinant of matrix \(A\) can be either \(+8\) or \(-8\).