Question

If \(A\) is a \(3 \times 3\) matrix and \( |A| = 5 \), find the value of \( |adj(A)| \).

• A. \(5\)
• B. \(25\)
• C. \(125\)
• D. \(15\)

Hint:

For an \(n \times n\) matrix, always remember the identity \( |adj(A)| = |A|^{n-1} \). For example, if the matrix is \(3 \times 3\), then \( |adj(A)| = |A|^2 \).

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
The problem asks for 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:
The relationship between the determinant of a matrix and the determinant of its adjoint is given by the formula:
\[ |adj(A)| = |A|^{n-1} \]
where \( n \) is the order of the square matrix \( A \).
Step 3: Detailed Explanation:
From the question, we have:
The determinant of matrix \( A \), \( |A| = 5 \).
The order of the matrix, \( n = 3 \).
Applying the formula:
\[ |adj(A)| = 5^{3-1} \]
\[ |adj(A)| = 5^2 \]
\[ |adj(A)| = 25 \]
Step 4: Final Answer:
The value of \( |adj(A)| \) is \( 25 \).