GATE EE 2016 Set 1 — Question 38
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Engineering Mathematics → Linear Algebra → Eigenvalues & Eigenvectors
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Question
Let the eigenvalues of a matrix be 1, -2 with eigenvectors and respectively. Then the eigenvalues and eigenvectors of the matrix would, respectively, be
Correct answer
(A) 2, 14; x₁, x₂
Solution
If is an eigenvalue of matrix with corresponding eigenvector , then for any polynomial function , the eigenvalue is with the same eigenvector .
Given eigenvalues of are and with eigenvectors and .
We need to find the eigenvalues of .
For :
. The eigenvector is .
For :
. The eigenvector is .
Thus, the eigenvalues are 2, 14 and the eigenvectors are .
Given eigenvalues of are and with eigenvectors and .
We need to find the eigenvalues of .
For :
. The eigenvector is .
For :
. The eigenvector is .
Thus, the eigenvalues are 2, 14 and the eigenvectors are .
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