Let $A$ be a $3 \times 3$ real matrix with $\det(A) = 24$. The determinant of the scalar multiplied matrix $23A$ is:
Options:
(A) 292008
(B) 552
(C) 12696
(D) 6716184
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 $n = 3$, $k = 23$, and $\det(A) = 24$:
$\det(23A) = 23^3 \times 24 = 12167 \times 24 = 292008$
Hence, the correct determinant value is $292008$.