GATE CS 2015 Set 2 — Question 15
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Engineering Mathematics → Linear Algebra → Eigenvalues & Eigenvectors
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Question
The larger of the two eigenvalues of the matrix is _________
Correct answer
6 to 6
Solution
Let the given matrix be .
To find the eigenvalues, we need to solve the characteristic equation , where is the identity matrix and represents the eigenvalues.Now, calculate the determinant:
This is a quadratic equation. We can solve it by factoring or using the quadratic formula.
Factoring the quadratic equation:
We need two numbers that multiply to -6 and add to -5. These numbers are -6 and 1.
This gives us two eigenvalues:
The eigenvalues are and .
The question asks for the larger of the two eigenvalues.
Comparing and , the larger eigenvalue is .Alternatively, using the trace and determinant:
For a matrix , the characteristic equation is .
Here, .
Trace .
Determinant .
So, the characteristic equation is , which is the same as obtained above.
Solving yields and .
The larger eigenvalue is .
To find the eigenvalues, we need to solve the characteristic equation , where is the identity matrix and represents the eigenvalues.Now, calculate the determinant:
This is a quadratic equation. We can solve it by factoring or using the quadratic formula.
Factoring the quadratic equation:
We need two numbers that multiply to -6 and add to -5. These numbers are -6 and 1.
This gives us two eigenvalues:
The eigenvalues are and .
The question asks for the larger of the two eigenvalues.
Comparing and , the larger eigenvalue is .Alternatively, using the trace and determinant:
For a matrix , the characteristic equation is .
Here, .
Trace .
Determinant .
So, the characteristic equation is , which is the same as obtained above.
Solving yields and .
The larger eigenvalue is .
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