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GATE 2014 EE (EE2) – Question 11

Engineering Mathematics · Linear Algebra: Eigen values, Eigen vectors · 1 mark · Multiple choice

Which statement holds for every real symmetric matrix?

  1. All eigenvalues are real.
  2. All eigenvalues are positive.
  3. All eigenvalues are distinct.
  4. The sum of eigenvalues is zero.

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Show answer and explanation

Correct answer: (A) All eigenvalues are real.

Explanation

The spectral theorem diagonalizes every real symmetric matrix by an orthogonal matrix, with a real diagonal. Thus **all eigenvalues are real,A**. They may be negative, repeated or have nonzero sum:the identity matrix already disproves distinctness and zero trace.