Let $A$ be a $3 \times 3$ real matrix with $\det(A) = 15$. The determinant of the scalar multiplied matrix $14A$ is:
Options:
(A) 41160
(B) 210
(C) 2940
(D) 576240
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 $n = 3$, $k = 14$, and $\det(A) = 15$:
$\det(14A) = 14^3 \times 15 = 2744 \times 15 = 41160$
Hence, the correct determinant value is $41160$.