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GATE 2025 ST – Question 48

Matrix Theory · Eigenvalues and Matrix Decompositions · 2 marks · Multiple select

Let O be all real 3×3 orthogonal matrices with determinant 1. Which hold?

  1. Some P∈O has eigenvalue 1/2.
  2. Some P∈O has eigenvalue 2.
  3. If λ is the only real eigenvalue of P∈O, then λ=1.
  4. Some P∈O has eigenvalue −1.

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Correct answer: (C) If λ is the only real eigenvalue of P∈O, then λ=1.; (D) Some P∈O has eigenvalue −1.

Explanation

Orthogonal eigenvalues have modulus 1, excluding A,B. A real 3×3 matrix has a real eigenvalue; if its other two eigenvalues are nonreal conjugates, their product is 1, so determinant 1 forces the real eigenvalue to be 1. For D, diag(1,−1,−1) belongs to O. **C,D**.

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