Question

Let \( A \) be a \(3 \times 3\) matrix and let \( B = 3A \). If \( |A| = 5 \), then the value of \( \frac{|\text{adj } B|}{|3A|} \) is equal to

• A. \(27 \)
• B. \(125 \)
• C. \(25 \)
• D. \(135 \)
• E. \(81 \)

Hint:

For adjoint: \( |\text{adj }A| = |A|^{n-1} \). For scalar multiple: \( |kA| = k^n |A| \).

Answer 1

The Correct Option is D

Concept:
• \( |kA| = k^n |A| \) for \( n \times n \) matrix
• \( |\text{adj }A| = |A|^{n-1} \)

Step 1: Find \( |B| \).
\[ B = 3A \Rightarrow |B| = 3^3 |A| = 27 \cdot 5 = 135 \]

Step 2: Find \( |\text{adj }B| \).
\[ |\text{adj }B| = |B|^{2} = 135^2 \]

Step 3: Find denominator.
\[ |3A| = 3^3 |A| = 27 \cdot 5 = 135 \]

Step 4: Compute value.
\[ \frac{|\text{adj }B|}{|3A|} = \frac{135^2}{135} = 135 \]