Linear Algebra

Recall $A \cdot \operatorname{adj}(A) = \det(A) \cdot I \implies \operatorname{adj}(A) = 4 A^{-1}$. Thus $2 A^{-1} \operatorname{adj}(A) = 2 A^{-1} (4 A^{-1}) = 8 (A^{-1})^2$. For a $3 \times 3$ matrix: $\det(8 (A^{-1})^2) = 8^3 [\det(A^{-1})]^2 = 512 \times \left(\frac{1}{4}\right)^2 = \frac{512}{16} = 32$ Hence, the determinant is **32**.