Allied & ICAR Exam Bank • ALLIED_0005 MCQ • 1 Mark

Section 1: Engineering Mathematics

Linear Algebra

Subtopic: Linear and orthogonal transformations

Question Statement:

Which of the following statements is TRUE for any real orthogonal matrix $Q$?

Options:

  • (A) $Q^T = Q^{-1}$ and $\det(Q) = \pm 1$
  • (B) $Q^T = -Q$ and $\det(Q) = 0$
  • (C) All eigenvalues of $Q$ must be zero
  • (D) $\text{Trace}(Q) = 0$ always

Answer Key:

A

Step-by-Step Solution & Explanation

An orthogonal matrix satisfies $Q^T Q = I$, meaning $Q^T = Q^{-1}$ and $(\det Q)^2 = 1 \implies \det Q = \pm 1$.
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