GATE CS 2014 Set 3 — Question 14
Go beyond PYQs with Success TrackerAI-powered personalised practice and doubt support. Unlimited practice on eligible plans; AI usage limits apply.MCQ+1 / -0.33EasyEigenvalues & EigenvectorsLinear AlgebraEngineering Mathematics
Engineering Mathematics → Linear Algebra → Eigenvalues & Eigenvectors
Last updated
Question
Which one of the following statements is TRUE about every matrix with only real eigenvalues?
Correct answer
(A) If the trace of the matrix is positive and the determinant of the matrix is negative, at least one of its eigenvalues is negative.
Solution
Let be the real eigenvalues of an matrix.
1.The determinant of a matrix is equal to the product of its eigenvalues: .
2.If the determinant is negative (), then the product of the eigenvalues is negative. For a product of real numbers to be negative, there must be at least one negative number in the set. Therefore, at least one eigenvalue must be negative. This makes statement (A) true.
Counter-examples for other options:- (B) Consider eigenvalues . The trace is , but one eigenvalue is negative.
- (C) Consider eigenvalues . The determinant is , but both eigenvalues are negative.
- (D) Consider eigenvalues . The trace is and the determinant is . Their product is , but two eigenvalues are negative.
Continue learning with Success Tracker
A step still unclear? Work through it with support
Use Success Tracker to ask about the reasoning, then try another GATE CS question to check your understanding.
AI-powered practice· Unlimited practice on eligible plans
- PYQs with solutions
- Attempt available previous-year questions, then compare your reasoning with the worked solution. Coverage varies by stream.
- Practice that adapts
- Choose a topic, work on weaker areas and bookmark questions to revisit. Your attempts feed your progress tracking.
- AI doubt support
- Ask follow-up questions about a step or concept while practising, instead of stopping at the final answer.
Unlimited practice is available on eligible plans. Free practice and AI usage have limits; check the current plan allowances before choosing.
This page stays readable without an account. AI responses can be wrong; check them against the solution and source material.
More questions on Linear Algebra
2026 Set 2 Q3A day can only be cloudy or sunny. The probability of a day being cloudy is , independent of…2026 Set 2 Q5‘When it is raining, peacocks dance.’ Based only on this sentence, which one of the following…2026 Set 2 Q8Figures (i) and (ii) represent intercity highway systems. The black dots represent cities and the…2026 Set 1 Q10An unbiased six-faced dice whose faces are marked with numbers 1, 2, 3, 4, 5, and 6 is rolled twice…2026 Set 2 Q10An unbiased six-faced dice whose faces are marked with numbers 1, 2, 3, 4, 5, and 6 is rolled twice…