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

Step 1: Understanding the Concept:
The determinant of an adjoint matrix is a function of the determinant of the original square matrix and its order $n$.
Step 2: Key Formula or Approach:
For any square matrix $A$ of order $n$: \[ |adj(A)| = |A|^{n-1} \]
Step 3: Detailed Explanation:
Given that $A$ is a $3 \times 3$ matrix, $n = 3$. We are given $|A| = 5$. Plugging these values into 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.