GATE EC 2022 Set 1 — Question 37
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Engineering Mathematics → Linear Algebra → Eigenvalues & Eigenvectors
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Question
Let be two non-zero real numbers and be two non-zero real vectors of size . Suppose that and satisfy , and . Let be the matrix given by:The eigenvalues of are __________.
Correct answer
(A) 0, α, β
Solution
Given . Consider the action of on :
.
Since and , we have .
Thus, is an eigenvalue of with eigenvector .Similarly, consider the action of on :
.
Thus, is an eigenvalue of with eigenvector .Since is a matrix and is the sum of two rank-1 matrices, its rank is at most 2. For a matrix with rank , at least one eigenvalue must be 0. Therefore, the three eigenvalues of are .
.
Since and , we have .
Thus, is an eigenvalue of with eigenvector .Similarly, consider the action of on :
.
Thus, is an eigenvalue of with eigenvector .Since is a matrix and is the sum of two rank-1 matrices, its rank is at most 2. For a matrix with rank , at least one eigenvalue must be 0. Therefore, the three eigenvalues of are .
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