Question Bank • QB_PARAMETRIC_EM_MAT_124MCQ • 1 Mark
Section 1: Engineering Mathematics
Linear Algebra: Matrices & Determinants
Subtopic: Determinant properties
Question Statement:
Let $A$ be a $3 \times 3$ real matrix with $\det(A) = 126$. The determinant of the scalar multiplied matrix $125A$ is: (Variation QB_PARAMETRIC_EM_MAT_124)
Options:
(A) 246093750
(B) 15750
(C) 1968750
(D) 30761718750
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 $n = 3$, $k = 125$, and $\det(A) = 126$:
$\det(125A) = 125^3 \times 126 = 1953125 \times 126 = 246093750$
Hence, the correct determinant value is $246093750$.