Let $A$ be a $3 \times 3$ real matrix with $\det(A) = 3$. The determinant of the scalar multiplied matrix $2A$ is:
Options:
(A) 24
(B) 6
(C) 12
(D) 48
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 $n = 3$, $k = 2$, and $\det(A) = 3$:
$\det(2A) = 2^3 \times 3 = 8 \times 3 = 24$
Hence, the correct determinant value is $24$.