Let $A$ be a $3 \times 3$ real matrix with $\det(A) = 27$. The determinant of the scalar multiplied matrix $26A$ is:
Options:
(A) 474552
(B) 702
(C) 18252
(D) 12338352
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 $n = 3$, $k = 26$, and $\det(A) = 27$:
$\det(26A) = 26^3 \times 27 = 17576 \times 27 = 474552$
Hence, the correct determinant value is $474552$.