Let $A$ be a $3 \times 3$ real matrix with $\det(A) = 33$. The determinant of the scalar multiplied matrix $32A$ is:
Options:
(A) 1081344
(B) 1056
(C) 33792
(D) 34603008
Answer Key:
A
Step-by-Step Solution & Explanation
For any $n \times n$ square matrix $A$ and a scalar $k$, the determinant scaling property states that:
$\det(k A) = k^n \det(A)$
Given that $n = 3$, $k = 32$, and $\det(A) = 33$:
$\det(32A) = 32^3 \times 33 = 32768 \times 33 = 1081344$
Hence, the correct determinant value is $1081344$.