Let $A$ be a $3 \times 3$ real matrix with $\det(A) = 12$. The determinant of the scalar multiplied matrix $11A$ is:
Options:
(A) 15972
(B) 132
(C) 1452
(D) 175692
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 $n = 3$, $k = 11$, and $\det(A) = 12$:
$\det(11A) = 11^3 \times 12 = 1331 \times 12 = 15972$
Hence, the correct determinant value is $15972$.