Let $A$ be a $3 \times 3$ real matrix with $\det(A) = 21$. The determinant of the scalar multiplied matrix $20A$ is:
Options:
(A) 168000
(B) 420
(C) 8400
(D) 3360000
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 $n = 3$, $k = 20$, and $\det(A) = 21$:
$\det(20A) = 20^3 \times 21 = 8000 \times 21 = 168000$
Hence, the correct determinant value is $168000$.