GATE CS 2015 Set 3 — Question 23
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Engineering Mathematics → Linear Algebra → Eigenvalues & Eigenvectors
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Question
In the given matrixone of the eigenvalues is 1. The eigenvectors corresponding to the eigenvalue 1 are
Correct answer
(B) \α(-4,2,1) α ≠ 0, α ∈ R\
Solution
Let the given matrix be .
We are given that is an eigenvalue.
To find the eigenvectors corresponding to , we need to solve the equation , where is the identity matrix and is the eigenvector.Substitute :
Let the eigenvector be .
The system of equations becomes:
1)
2) (This equation is trivial and provides no new information)
3) Now, substitute from equation (1) into equation (3):
So, the eigenvector can be written as:
We can factor out :
Since an eigenvector must be non-zero, . Let , where is any non-zero real number.
Thus, the set of eigenvectors corresponding to the eigenvalue 1 is .This matches option (B).
We are given that is an eigenvalue.
To find the eigenvectors corresponding to , we need to solve the equation , where is the identity matrix and is the eigenvector.Substitute :
Let the eigenvector be .
The system of equations becomes:
1)
2) (This equation is trivial and provides no new information)
3) Now, substitute from equation (1) into equation (3):
So, the eigenvector can be written as:
We can factor out :
Since an eigenvector must be non-zero, . Let , where is any non-zero real number.
Thus, the set of eigenvectors corresponding to the eigenvalue 1 is .This matches option (B).
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