GATE EC 2024 Set 1 — Question 55
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Engineering Mathematics → Linear Algebra → Eigenvalues & Eigenvectors
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Question
Consider the matrix , where is a positive real number. Which of the following vectors is/are eigenvector(s) of this matrix?
Correct answer
(A) bmatrix 1 \ -√(2/k) bmatrix; (B) bmatrix 1 \ √(2/k) bmatrix
Solution
The characteristic equation of the matrix is given by:
.For :
.
If , then , which matches option (B).For :
.
If , then , which matches option (A).Checking (C) and (D): If , then . Thus, is not an eigenvector.
.For :
.
If , then , which matches option (B).For :
.
If , then , which matches option (A).Checking (C) and (D): If , then . Thus, is not an eigenvector.
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