Question

If \( A = \begin{bmatrix} -2 & 0 & 0 \\ 1 & 2 & 3 \\ 5 & 1 & -1 \end{bmatrix} \), then the value of \( |A \cdot \text{adj}(A)| \) is:

• A. \( 100 \) I
• B. \( 10 \) I
• C. \( 10 \)
• D. \( 1000 \)

Hint:

To compute \( |A \cdot \text{adj}(A)| \), use the property \( |A \cdot \text{adj}(A)| = |A|^n \) for \( n \times n \) matrices.

Answer 1

The Correct Option is D

Step 1: Determinant Property
For any square matrix \( A \) of size \( n \), the determinant of the product of \( A \) and its adjugate matrix, \( |A \cdot \text{adj}(A)| \), is equal to \( |A|^n \).

Step 2: Determinant Calculation
Using cofactor expansion, compute the determinant of \( A \): \[ |A| = -2 \cdot \begin{vmatrix} 2 & 3 \\ 1 & -1 \end{vmatrix} = -2((-2) - 3) = 10. \]
Step 3: Calculate \( |A \cdot \text{adj}(A)| \)
Given that the matrix size \( n = 3 \): \[ |A \cdot \text{adj}(A)| = |A|^3 = 10^3 = 1000. \]
Step 4: Option Verification
The calculated value is \( 1000 \), which matches option (D).