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GATE 2025 BM – Question 21

Engineering Mathematics · Linear Algebra · 1 mark · Multiple choice

Find the eigenvalues of $\begin{pmatrix}\cos\theta&-\sin\theta\\\sin\theta&\cos\theta\end{pmatrix}$.

  1. $\cos\theta,\sin\theta$
  2. $e^{i\theta},e^{-i\theta}$
  3. $\cos\theta,-\sin\theta$
  4. $\cos^2\theta,\sin^2\theta$

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Correct answer: (B) $e^{i\theta},e^{-i\theta}$

Explanation

The characteristic polynomial is λ²−2cosθ λ+1. Its roots are cosθ±i sinθ, which Euler’s formula identifies as eⁱᶿ and e⁻ⁱᶿ. Thus B; a planar rotation need not have real eigenvalues.

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