### Related Formula sum_k a_ik A_jk = delta_ij |A| implies C = beginbmatrix |A| & 0 \\ 0 & |A| endbmatrix implies |C| = |A|^2$\sum_{k} a_{ik} A_{jk} = \delta_{ij} |A| \implies C = \begin{\bmatrix} |A| & 0 \\ 0 & |A| \end{\bmatrix} \implies |C| = |A|^2$ ### Core Logic Evaluate the determinant of matrix A$A$: |A| = (log_5 128)(log_4 25) - (log_4 5)(log_5 8)$|A| = (\log_5 128)(\log_4 25) - (\log_4 5)(\log_5 8)$ Using change of base rules: |A| = left(7log_5 2right)left(2log_4 5right) - left(frac12log_2 5right)left(3log_5 2right)$|A| = \left(7\log_5 2\right)\left(2\log_4 5\right) - \left(\frac{1}{2}\log_2 5\right)\left(3\log_5 2\right)$ |A| = 14left(log_5 2 cdot frac12log_2 5right) - frac32 = 7 - 1.5 = 5.5 = frac112$|A| = 14\left(\log_5 2 \cdot \frac{1}{2}\log_2 5\right) - \frac{3}{2} = 7 - 1.5 = 5.5 = \frac{11}{2}$ ### Step 1: Compute |C| and evaluate response target Since matrix properties dictate |C| = |A|^2$|C| = |A|^2$: |C| = left(frac112right)^2 = frac1214$|C| = \left(\frac{11}{2}\right)^2 = \frac{121}{4}$ Evaluate targeted multiplier: 8|C| = 8 times frac1214 = 2 times 121 = 242$8|C| = 8 \times \frac{121}{4} = 2 \times 121 = 242$ ### Pattern Recognition Recognize the core cofactor theorem identity instantly: multiplying rows by cofactors of other rows creates zero elements, yielding basic diagonal scalar structures matching matrix attributes. ### Evaluation Rubric / Model Answer null ### Chapter Mix Class 12 Mathematics: Matrices and Determinants