The GATE Grind

GATE 2023 MN – Question 27

Engineering Mathematics · Linear Algebra · 1 mark · Multiple choice

Matrix A is negative definite. Which statement is NOT correct? Use the conventional real symmetric matrix definition.

  1. A is symmetric.
  2. The determinant of A is always negative.
  3. All eigenvalues are negative.
  4. The trace of A is always negative.

Practise this question in The GATE Grind →

Show answer and explanation

Correct answer: (B) The determinant of A is always negative.

Explanation

Negative definiteness of a real symmetric matrix means every eigenvalue is negative. The trace, their sum, is negative. The determinant is their product and has sign (−1)ⁿ, so it is positive for even order. Thus “always negative” is false, B. Symmetry is part of the convention intended by this question.

More Linear Algebra questions

All Linear Algebra questions →