The GATE Grind

GATE 2014 CS (CS3) – Question 14

Engineering Mathematics · Linear Algebra · 1 mark · Multiple choice

Which statement is TRUE for every n × n matrix having only real eigenvalues?

  1. Positive trace and negative determinant imply at least one negative eigenvalue.
  2. Positive trace implies all eigenvalues are positive.
  3. Positive determinant implies all eigenvalues are positive.
  4. A positive product of trace and determinant implies all eigenvalues are positive.

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Show answer and explanation

Correct answer: (A) Positive trace and negative determinant imply at least one negative eigenvalue.

Explanation

The determinant is the product of the real eigenvalues. A negative determinant requires an odd number of negative factors, so at least one eigenvalue is negative. Thus **A** is true; the positive trace does not force all eigenvalues positive. Zero eigenvalues are excluded by the negative determinant.