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: Understanding the Question:
The question asks for the ratio of the determinant of the adjoint of matrix \( B \) to the determinant of matrix \( C \), where \( B \) is the adjoint of \( A \) and \( C \) is \( 5A \).
Step 2: Key Formula or Approach:
For a square matrix \( A \) of order \( n = 3 \):
1. \( |A| = \det(A) \)
2. \( |\text{adj } A| = |A|^{n-1} = |A|^2 \)
3. \( \text{adj } B = \text{adj}(\text{adj } A) \implies |\text{adj } B| = |A|^{(n-1)^2} = |A|^4 \)
4. \( |kA| = k^n |A| = 5^3 |A| = 125|A| \)
Step 3: Detailed Explanation:
First, calculate the determinant of \( A \):
\[ |A| = 1(0 - (-3)) - (-1)(0 - (-6)) + 1(0 - 4) \] \[ |A| = 1(3) + 1(6) - 4 = 3 + 6 - 4 = 5 \] Now, find \( |\text{adj } B| \):
\[ |\text{adj } B| = |\text{adj}(\text{adj } A)| = |A|^4 = 5^4 = 625 \] Next, find \( |C| \):
\[ |C| = |5A| = 5^3 \times |A| = 125 \times 5 = 625 \] Calculate the ratio:
\[ \frac{|\text{adj } B|}{|C|} = \frac{625}{625} = 1 \] Step 4: Final Answer:
The ratio is 1.