GATE DA 2025 Set 1 — Question 38
Go beyond PYQs with Success TrackerAI-powered personalised practice and doubt support. Unlimited practice on eligible plans; AI usage limits apply.MCQ+2 / -0.67MediumLinear Independence & DependenceVector Spaces, Basis & DimensionsLinear AlgebraPositive Definite MatricesEigenvalues & EigenvectorsInverse of a MatrixSystems of Linear Equations
Linear Algebra → Eigenvalues & Eigenvectors → Inverse of a Matrix
Last updated
Question
Let be a set of linearly independent vectors in . Let the -th element of matrix be given by , . Which one of the following statements is correct?
Correct answer
(A) A is invertible
Solution
Let be the matrix whose rows are . Since , is an matrix.
The entry corresponds to the dot product of the -th row and -th row (if we consider them as vectors). More precisely, if we form the matrix with rows , then the matrix product has entries .
Thus, .
Since the set is linearly independent in , the rows of are linearly independent. Since is square (), it has full rank and is invertible ().
Therefore, .
Since , is invertible.(B) Since is invertible, 0 cannot be a singular value.
(C) .
(D) is a Gram matrix of linearly independent vectors, so it is symmetric positive definite. Thus for all non-zero .
The entry corresponds to the dot product of the -th row and -th row (if we consider them as vectors). More precisely, if we form the matrix with rows , then the matrix product has entries .
Thus, .
Since the set is linearly independent in , the rows of are linearly independent. Since is square (), it has full rank and is invertible ().
Therefore, .
Since , is invertible.(B) Since is invertible, 0 cannot be a singular value.
(C) .
(D) is a Gram matrix of linearly independent vectors, so it is symmetric positive definite. Thus for all non-zero .
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 DA 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.