Allied & ICAR Exam Bank • ALLIED_1013 MCQ • 1 Mark

Section 1: Engineering Mathematics

Linear Algebra

Subtopic: Eigen values and Eigen vectors

Question Statement:

If an orthogonal matrix $Q$ has real entries, which of the following statements is ALWAYS TRUE regarding its determinant?

Options:

  • (A) \det(Q) = \pm 1
  • (B) \det(Q) = 0
  • (C) \det(Q) = 2
  • (D) \det(Q) > 1

Answer Key:

A

Step-by-Step Solution & Explanation

For an orthogonal matrix $Q$, $Q^T Q = I$. Taking determinants on both sides: $\det(Q^T) \det(Q) = (\det(Q))^2 = \det(I) = 1 \implies \det(Q) = \pm 1$.
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