Question

Let \[ A = \left[ a_{ij} \right] = \begin{bmatrix} \log_5 128 & \log_4 5 \\ \log_5 8 & \log_4 25 \end{bmatrix}. \] 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. 242
• B. 222
• C. 262
• D. 288

Hint:

When calculating the determinant of a matrix, use the cofactor expansion method and carefully compute the matrix's minors and cofactors to get the correct result.

Answer 1

The Correct Option is C

Step 1: First, calculate the determinant of matrix \( A \), which involves calculating the cofactors for each element and the matrix itself. 
Step 2: Use the formula for the cofactor matrix \( C \), and evaluate the determinant of \( C \) by applying the cofactor expansion. 
Step 3: After calculating the determinant, multiply the result by 8 to get \( 8|C| \), which evaluates to 262. Thus, the correct answer is (3).