Allied & ICAR Exam Bank • ALLIED_2607MSQ • 2 Marks
Section 1: Engineering Mathematics
Linear Algebra
Subtopic: Matrices and determinants
Question Statement:
Which of the following statements are TRUE for any real square matrices $A$ and $B$ of order $n$?
Options:
(A) $\det(A B) = \det(A) \det(B)$
(B) $\det(A + B) = \det(A) + \det(B)$
(C) $\det(A^T) = \det(A)$
(D) $\text{trace}(A B) = \text{trace}(B A)$
Answer Key:
A, C, D
Step-by-Step Solution & Explanation
Determinant is multiplicative ($\det(AB) = \det(A)\det(B)$) and invariant under transpose. Trace is cyclic ($\text{tr}(AB) = \text{tr}(BA)$). However, determinant is generally NOT additive ($\det(A+B) \neq \det(A) + \det(B)$).