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GATE 2014 EC (EC3) – Question 37

Engineering Mathematics · Linear Algebra · 2 marks · Multiple choice

Which statement is NOT true for a square matrix A?

  1. If A is upper triangular, its eigenvalues are its diagonal elements.
  2. If A is real symmetric, its eigenvalues are always real and positive.
  3. If A is real, A and its transpose always have the same eigenvalues.
  4. If every principal minor of A is positive, every eigenvalue of A is positive.

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Show answer and explanation

Correct answer: (B) If A is real symmetric, its eigenvalues are always real and positive.

Explanation

A real symmetric matrix has real eigenvalues, but they need not be positive. For example−Iis real symmetric and has eigenvalues−1. Thus **B** is the officially keyed false statement. There is also a missing hypothesis in D: positive principal minors guarantee positive eigenvalues for symmetric matrices, but a general nonsymmetric P-matrix may have nonreal eigenvalues, e.g. [[1,−1],[1,1]] has positive principal minors and eigenvalues1±j.