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GATE 2020 ST – Question 37

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

Over C³, $T(x,y,z)=(x,\sqrt3y/2-z/2,y/2+\sqrt3z/2)$. Which is true?

  1. A nonzero X satisfies T(X)=−X.
  2. A nonzero Y has real eigenvalue λ≠1.
  3. T is diagonalizable.
  4. T²=I₃.

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Show answer and explanation

Correct answer: (C) T is diagonalizable.

Explanation

The y,z block is a30° rotation. Its complex eigenvalues are√3/2±i/2, while the x-axis eigenvalue is1. These three distinct eigenvalues give a complex eigenbasis, so C. None is−1, the only real eigenvalue is1, and squaring the rotation gives60°, not identity.

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