Question Bank • QB_PARAMETRIC_EM_MAT_115MCQ • 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 $\det(A) = 117$. The determinant of the scalar multiplied matrix $116A$ is: (Variation QB_PARAMETRIC_EM_MAT_115)
Options:
(A) 182624832
(B) 13572
(C) 1574352
(D) 21184480512
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 $n = 3$, $k = 116$, and $\det(A) = 117$:
$\det(116A) = 116^3 \times 117 = 1560896 \times 117 = 182624832$
Hence, the correct determinant value is $182624832$.