Question

If \(A\) is a square matrix of order \(3\) and \(|A| = 5\), find \(|adj(A)|\).

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

Hint:

Always remember the identity: \[ |adj(A)| = |A|^{n-1} \] where \(n\) is the order of the square matrix. This formula helps quickly evaluate determinants of adjoint matrices.

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
The question requires calculating the determinant of the adjoint of a matrix \(A\), given the determinant of the matrix itself and its order.
Step 2: Key Formula or Approach:
For any square matrix \(A\) of order \(n\), the relationship between the determinant of the matrix and its adjoint is:
\[ |adj(A)| = |A|^{n-1} \]
Step 3: Detailed Explanation:
Given:
Order of the matrix, \(n = 3\)
Determinant of the matrix, \(|A| = 5\)
Using the formula:
\[ |adj(A)| = 5^{3-1} \]
\[ |adj(A)| = 5^2 \]
\[ |adj(A)| = 25 \]
Step 4: Final Answer:
The value of \(|adj(A)|\) is \(25\).