GATE DA 2025 Set 1 — Question 28
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Linear Algebra → Eigenvalues & Eigenvectors → Eigenvalue Computation
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Question
Let , where is the identity matrix and , . Which of the following options is/are correct?
Correct answer
(A) Rank of A is n; (B) A is invertible
Solution
Given with .The eigenvalues of the rank-1 matrix are:
(A) Since all eigenvalues are non-zero (2 and 1), the matrix has full rank . Correct.
(B) Since the determinant (product of eigenvalues) is non-zero, is invertible. Correct.
(C) 0 is not an eigenvalue. Incorrect.
(D) The eigenvalues of are the reciprocals of the eigenvalues of , i.e., and . None are negative. Incorrect.
- (with multiplicity 1, eigenvector )
- (with multiplicity , eigenvectors orthogonal to )
(A) Since all eigenvalues are non-zero (2 and 1), the matrix has full rank . Correct.
(B) Since the determinant (product of eigenvalues) is non-zero, is invertible. Correct.
(C) 0 is not an eigenvalue. Incorrect.
(D) The eigenvalues of are the reciprocals of the eigenvalues of , i.e., and . None are negative. Incorrect.
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