GATE EE 2026 Set 1 — Question 30
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Engineering Mathematics → Linear Algebra → Eigenvalues & Eigenvectors
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Question
Two matrices and have a common eigenvalue , and the same corresponding nonzero eigenvector.Which of the following options is/are correct?(Note: is the identity matrix.)
Correct answer
(A) Determinant (A - 2I) = 0; (B) Determinant (B - 2I) = 0; (D) Determinant (A + B - 4I) = 0
Solution
Let be the common eigenvector such that and .
- Option (A): Since is an eigenvalue of , the matrix is singular, meaning its determinant is zero. Thus, is correct.
- Option (B): Similarly, since is an eigenvalue of , is correct.
- Option (C): Consider . Since , is not in the null space of . This does not guarantee that the determinant is zero.
- Option (D): Consider . Since there exists a non-zero vector such that , the matrix is singular, and its determinant must be zero. Thus, is correct.
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