Let $A$ be a $3 \times 3$ real matrix with $\det(A) = 9$. The determinant of the scalar multiplied matrix $8A$ is:
Options:
(A) 4608
(B) 72
(C) 576
(D) 36864
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 $n = 3$, $k = 8$, and $\det(A) = 9$:
$\det(8A) = 8^3 \times 9 = 512 \times 9 = 4608$
Hence, the correct determinant value is $4608$.