Let $A$ be a $3 \times 3$ real matrix with $\det(A) = 36$. The determinant of the scalar multiplied matrix $35A$ is:
Options:
(A) 1543500
(B) 1260
(C) 44100
(D) 54022500
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 $n = 3$, $k = 35$, and $\det(A) = 36$:
$\det(35A) = 35^3 \times 36 = 42875 \times 36 = 1543500$
Hence, the correct determinant value is $1543500$.