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: Property of determinants
For a square matrix \( A \), \( |A \cdot \text{adj}(A)| = |A|^n \), where \( n \) is the size of \( A \).

Step 2: Compute \( |A| \)
Using cofactor expansion: \[ |A| = -2 \cdot \begin{vmatrix} 2 & 3 \\ 1 & -1 \end{vmatrix} = -2((-2) - 3) = 10. \]
Step 3: Calculate \( |A \cdot \text{adj}(A)| \)
Since \( n = 3 \): \[ |A \cdot \text{adj}(A)| = |A|^3 = 10^3 = 1000. \]
Step 4: Verify the options
The correct value is \( 1000 \), which corresponds to option (D).