GATE CE 2023 Set 1 — Question 47
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Engineering Mathematics → Linear Algebra → Eigenvalues & Eigenvectors
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Question
For the matrix
which of the following statements is/are TRUE?
which of the following statements is/are TRUE?
Correct answer
(A) The eigenvalues of [A]^T are same as the eigenvalues of [A]; (B) The eigenvalues of [A]⁻¹ are the reciprocals of the eigenvalues of [A]; (D) The eigenvectors of [A]⁻¹ are same as the eigenvectors of [A]
Solution
Statement (A): The characteristic equation is the same for and because . Thus, eigenvalues are identical. TRUE.
Statement (B): If , then , which implies . Since , is invertible and eigenvalues of are reciprocals of eigenvalues of . TRUE.
Statement (C): Eigenvectors of and are the same only if is a symmetric matrix. Here is not symmetric, so eigenvectors are generally different. FALSE.
Statement (D): As shown in (B), . The eigenvector remains the same for both and . TRUE.
Statement (B): If , then , which implies . Since , is invertible and eigenvalues of are reciprocals of eigenvalues of . TRUE.
Statement (C): Eigenvectors of and are the same only if is a symmetric matrix. Here is not symmetric, so eigenvectors are generally different. FALSE.
Statement (D): As shown in (B), . The eigenvector remains the same for both and . TRUE.
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