Question

If \( A = \begin{bmatrix} 1 & -1 & 1 \\ 0 & 2 & -3 \\ 2 & 1 & 0 \end{bmatrix} \), \( B = \text{adj } A \) and \( C = 5A \), then \( \frac{|\text{adj } B|}{|C|} = \)}

• A. 2
• B. 4
• C. 1
• D. 5

Hint:

$|\text{adj}(\text{adj } A)| = |A|^{(n-1)^2}$ and $|kA| = k^n |A|$.

Answer 1

The Correct Option is C

Step 1: Find $|A|$
$|A| = 1(0+3) - (-1)(0+6) + 1(0-4) = 3 + 6 - 4 = 5$.
Step 2: Evaluate $|\text{adj B|$}}
$B = \text{adj } A \implies |\text{adj } B| = |\text{adj}(\text{adj } A)|$.
Formula: $|\text{adj}(\text{adj } A)| = |A|^{(n-1)^2}$. For $n=3$, $|A|^4 = 5^4 = 625$.
Step 3: Evaluate $|C|$
$C = 5A \implies |C| = 5^n |A| = 5^3 \times 5 = 5^4 = 625$.
Step 4: Ratio
Ratio $= \frac{625}{625} = 1$.
Final Answer:(C)