Question

If \( A \) is a square matrix of order \( 2 \times 2 \) and \( |A| = 5 \), then \( | \text{Adj}(A) | \) is:

• A. 25
• B. 125
• C. 5
• D. 10

Hint:

For any \( n \times n \) matrix \( A \), \( |\text{Adj}(A)| = |A|^{n-1} \), which is useful for calculating the determinant of the adjugate matrix.

Answer 1

The Correct Option is A

Step 1: Understanding the concept of adjugate matrix.
For a square matrix \( A \) of order \( n \), the determinant of the adjugate matrix \( \text{Adj}(A) \) is related to the determinant of the matrix \( A \) by the following formula: \[ |\text{Adj}(A)| = |A|^{n-1} \] Since \( A \) is a \( 2 \times 2 \) matrix, we have \( n = 2 \). Therefore, \[ |\text{Adj}(A)| = |A|^{2-1} = |A| = 5 \] Step 2: Conclusion.
Thus, the determinant of the adjugate matrix is \( 25 \), corresponding to option (A).