Let $A$ be a $3 \times 3$ real matrix with $\det(A) = 6$. The determinant of the scalar multiplied matrix $5A$ is:
Options:
(A) 750
(B) 30
(C) 150
(D) 3750
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 $n = 3$, $k = 5$, and $\det(A) = 6$:
$\det(5A) = 5^3 \times 6 = 125 \times 6 = 750$
Hence, the correct determinant value is $750$.