Let $A$ be a $3 \times 3$ real matrix with $\det(A) = 30$. The determinant of the scalar multiplied matrix $29A$ is:
Options:
(A) 731670
(B) 870
(C) 25230
(D) 21218430
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 $n = 3$, $k = 29$, and $\det(A) = 30$:
$\det(29A) = 29^3 \times 30 = 24389 \times 30 = 731670$
Hence, the correct determinant value is $731670$.