Let $A$ be a $3 \times 3$ real matrix with $\det(A) = 18$. The determinant of the scalar multiplied matrix $17A$ is:
Options:
(A) 88434
(B) 306
(C) 5202
(D) 1503378
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 $n = 3$, $k = 17$, and $\det(A) = 18$:
$\det(17A) = 17^3 \times 18 = 4913 \times 18 = 88434$
Hence, the correct determinant value is $88434$.