Question

If \(A\) is a \(3\times3\) matrix such that \(|A| = 4\) and \(B = \text{adj}\,A\), find the value of \(|B|\).

• A. \(4\)
• B. \(8\)
• C. \(16\)
• D. \(64\)

Hint:

For an \(n \times n\) matrix: \[ |\text{adj}(A)| = |A|^{\,n-1} \] Special cases:
• If \(A\) is \(2\times2\): \(|\text{adj}(A)| = |A|\)
• If \(A\) is \(3\times3\): \(|\text{adj}(A)| = |A|^2\) This property is frequently used in determinant and inverse matrix problems.

Answer 1

The Correct Option is C

Topic: Determinants and Properties of Adjoint Matrices
Step 1: Understanding the Question:
The question asks for the determinant of the adjoint of a given matrix \(A\).
We are provided with the determinant of matrix \(A\) and its order.
Step 2: Key Formula or Approach:
For any square matrix \(A\) of order \(n \times n\), the relationship between the determinant of its adjoint and the determinant of the original matrix is given by:
\[ |\text{adj}(A)| = |A|^{n-1} \]
Step 3: Detailed Explanation:
1. Identify the given values:
Determinant of matrix \(A\), \(|A| = 4\).
Order of the matrix, \(n = 3\) (since it is a \(3 \times 3\) matrix).
2. Substitute the values into the formula:
\[ |B| = |\text{adj}(A)| = |A|^{3-1} \]
\[ |B| = 4^2 \]
3. Calculate the final result:
\[ |B| = 16 \]
Step 4: Final Answer:
The value of \(|B|\) is \(16\).