Allied & ICAR Exam Bank • ALLIED_2608 MCQ • 1 Mark

Section 1: Engineering Mathematics

Linear Algebra

Subtopic: Linear and orthogonal transformations

Question Statement:

Let $Q$ be a real orthogonal matrix ($Q^T Q = I$). Which of the following properties must ALWAYS hold?

Options:

  • (A) $\det(Q) = \pm 1$
  • (B) $\det(Q) = 0$
  • (C) All eigenvalues of $Q$ are real
  • (D) $Q = Q^T$

Answer Key:

A

Step-by-Step Solution & Explanation

Since $Q^T Q = I$, $\det(Q^T Q) = (\det(Q))^2 = 1 \implies \det(Q) = \pm 1$. Eigenvalues have magnitude 1 but may be complex conjugates.
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