Question

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

• A. \(25\)
• B. \(2\)
• C. \(1\)
• D. \(5\)

Hint:

Find |A| first, then use |adj A| = |A|^2 and |adj B| = |B|^2 for a 3 by 3 matrix, and |5A| = 5^3 |A|.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept
For a \(3\times 3\) matrix, \(|\text{adj}\,M|=|M|^{2}\) and \(|kM|=k^{3}|M|\).

Step 2: Find |A|
\(|A|=1(0+3)+1(0+6)+1(0-4)=3+6-4=5\).

Step 3: Find |adj B| and |C|
\(|B|=|\text{adj}\,A|=|A|^{2}=25\), so \(|\text{adj}\,B|=|B|^{2}=625\).
\(|C|=|5A|=125\times 5=625\).

Step 4: Take the ratio
\(\frac{|\text{adj}\,B|}{|C|}=\frac{625}{625}=1\).

Final Answer:
The required ratio is 1. \[ \boxed{\text{(C) }1} \]