Question

Let A = \(\begin{bmatrix} \log_5 128 & \log_4 5  \log_5 8 & \log_4 25 \end{bmatrix}\) \). If \(A_{ij}\)  is the cofactor of \( a_{ij} \), \( C_{ij} = \sum_{k=1}^2 a_{ik} A_{jk} \), and \( C = [C_{ij}] \), then \( 8|C| \) is equal to:

• A. 262
• B. 222
• C. 242
• D. 288

Hint:

To calculate the determinant of a matrix with cofactors, use the cofactor expansion method and simplify the resulting expression.

Answer 1

The Correct Option is C

We are tasked with analyzing the given expressions and determining the value of \( 8|C| \). Let us proceed step by step:

1. Determinant of Matrix \( A \):
The determinant of matrix \( A \) is given as:

\( |A| = \frac{11}{2} \)

2. Cofactor Expressions:
The cofactors \( C_{ij} \) are computed as follows:

• \( C_{11} = \sum_{k=1}^2 a_{1k} \cdot A_{1k} = a_{11}A_{11} + a_{12}A_{12} = |A| = \frac{11}{2} \)
• \( C_{12} = \sum_{k=1}^2 a_{1k} \cdot A_{2k} = 0 \)
• \( C_{21} = \sum_{k=1}^2 a_{2k} \cdot A_{1k} = 0 \)
• \( C_{22} = \sum_{k=1}^2 a_{2k} \cdot A_{2k} = |A| = \frac{11}{2} \)

3. Matrix \( C \):
The matrix \( C \) is constructed using the cofactor values:

\( C = \begin{bmatrix} \frac{11}{2} & 0 \\ 0 & \frac{11}{2} \end{bmatrix} \)

4. Determinant of Matrix \( C \):
The determinant of \( C \) is calculated as:

\( |C| = \left(\frac{11}{2}\right) \cdot \left(\frac{11}{2}\right) - (0 \cdot 0) = \frac{121}{4} \)

5. Scaling \( |C| \):
We are asked to compute \( 8|C| \):

\( 8|C| = 8 \cdot \frac{121}{4} = 2 \cdot 121 = 242 \)

Final Answer:
The value of \( 8|C| \) is \( \boxed{242} \).