GATE CE 2015 Set 1 — Question 46
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Engineering Mathematics → Linear Algebra → Eigenvalues & Eigenvectors
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Question
The smallest and largest Eigen values of the following matrix are:
Correct answer
(D) 1.0 and 2.0
Solution
To find the eigenvalues of the matrix , we can use the properties of eigenvalues:
1.Trace Property: The sum of the eigenvalues is equal to the trace of the matrix (sum of diagonal elements).
2.Determinant Property: The product of the eigenvalues is equal to the determinant of the matrix.
Now, let's check the options for the smallest and largest eigenvalues:- Option (A): 1.5 and 2.5. If these are the smallest and largest, the third eigenvalue must be between them. Sum = . Product = . (Incorrect)
- Option (B): 0.5 and 2.5. Sum = . Product = . (Incorrect)
- Option (C): 1.0 and 3.0. Sum = . Product = . (Incorrect)
- Option (D): 1.0 and 2.0. Sum = . Product = . (Correct)
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