Let $A$ be a $3 \times 3$ invertible real matrix with $\det(A) = 4$. The determinant of the matrix $2 \operatorname{adj}(A)$ is ________ (answer in integer).
Answer Key:
128
Step-by-Step Solution & Explanation
For an $n \times n$ matrix $A$ and a scalar $k$:
$\det(k B) = k^n \det(B)$
Here $B = \operatorname{adj}(A)$ and $n = 3$, so:
$\det(2 \operatorname{adj}(A)) = 2^3 \det(\operatorname{adj}(A)) = 8 \det(\operatorname{adj}(A))$
Using the standard identity $\det(\operatorname{adj}(A)) = (\det(A))^{n-1}$ for $n = 3$:
$\det(\operatorname{adj}(A)) = (\det(A))^{3-1} = 4^2 = 16$
Substituting this back:
$\det(2 \operatorname{adj}(A)) = 8 \times 16 = 128$