Question

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

• A. 25
• B. 125
• C. 5
• D. 10

Hint:

Be careful with the order. If the matrix was \( 3 \times 3 \), the answer would be \( |A|^{3-1} = 5^2 = 25 \). Always identify \( n \) first.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
There is a direct relationship between the determinant of a matrix and the determinant of its adjoint.
Step 2: Key Formula or Approach:
For any square matrix \( A \) of order \( n \times n \), the determinant of its adjoint is given by:
\[ |Adj(A)| = |A|^{n-1} \] Step 3: Detailed Explanation:
Given in the problem:
1. Order of the matrix, \( n = 2 \).
2. Determinant of the matrix, \( |A| = 5 \).
Substituting these values into the formula:
\[ |Adj(A)| = 5^{2-1} \] \[ |Adj(A)| = 5^1 \] \[ |Adj(A)| = 5 \] Step 4: Final Answer:
The value of \( |Adj(A)| \) is 5.