Question Bank • QB_EM_136 MCQ • 1 Mark

Section 1: Engineering Mathematics

Linear Algebra

Subtopic: Linear and orthogonal transformations

Question Statement:

If a real square matrix $Q$ is orthogonal ($Q^T Q = I$), then the determinant of $Q$ must be:

Options:

  • (A) +1 or -1
  • (B) 0
  • (C) +2 or -2
  • (D) Any arbitrary real number

Answer Key:

A

Step-by-Step Solution & Explanation

Since $Q^T Q = I$, taking determinants gives $\det(Q^T) \det(Q) = (\det(Q))^2 = \det(I) = 1$, which implies $\det(Q) = \pm 1$.
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