Question Bank • QB_SUB_EM_0003 MSQ • 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 matrix $A$?

Options:

  • (A) $\det(A) = \det(A^T)$
  • (B) The eigenvalues of $A$ and $A^T$ are identical
  • (C) If $A$ is orthogonal, $\det(A) = \pm 1$
  • (D) $\det(A + B) = \det(A) + \det(B)$ for all matrices $B$

Answer Key:

A, B, C

Step-by-Step Solution & Explanation

Determinants and spectra are invariant under transposition, and orthogonal matrices satisfy $A^T A = I \implies (\det A)^2 = 1 \implies \det A = \pm 1$. The determinant is not linear: $\det(A + B) \neq \det(A) + \det(B)$ in general.
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