Math Problem Statement
Question 8 select two eigenvectors of the matrix A= [[3/2,-1][-1/2,1/2]] . A. v1= [-1-sqrroot(5), 1], v2 = [-1+sqrroot(5), 1]. B. v1= [1-sqrroot(3), 1], v2 = [1+sqrroot(3), 1]. C. v1= [1-sqrroot(5), 1], v2 = [1+sqrroot(5), 1]. D. v1= [-1-sqrroot(3), 1], v2 = [-1+sqrroot(3), 1]
Solution
To determine which pair of vectors are eigenvectors of the matrix , we need to verify which vectors satisfy the eigenvector equation:
where is the eigenvector and is the corresponding eigenvalue.
Step 1: Find the eigenvalues
The eigenvalues can be found by solving the characteristic equation:
Where is the identity matrix. So, we need to compute:
Expanding this determinant:
Simplifying the expression:
Expanding the product:
This is a quadratic equation. The solutions for are:
So, the eigenvalues are:
Step 2: Verify which eigenvectors correspond to these eigenvalues
The next step is to plug in the candidate eigenvectors from each option into the equation to see if they satisfy the equation.
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Option A:
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Option B:
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Option C:
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Option D:
By substituting the vectors from each option into the equation , the correct answer will be the one that satisfies the equation for each eigenvalue.
Step 3: Conclusion
After testing each pair of vectors, the correct answer is Option C:
These vectors are eigenvectors corresponding to the eigenvalues found from the characteristic equation.
Would you like further details on the calculations, or do you have any other questions?
Relative Questions:
- How do you compute the characteristic equation of a matrix?
- Why are eigenvectors important in the context of linear transformations?
- What is the geometric interpretation of eigenvectors?
- How can eigenvalues be used in practical applications like stability analysis?
- What are the conditions under which a matrix can have complex eigenvalues?
Tip: Always double-check the signs when calculating eigenvalues and eigenvectors, as small errors in sign can lead to incorrect results.
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Eigenvalues
Eigenvectors
Matrix Operations
Formulas
Characteristic equation
Determinant calculation
Theorems
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Suitable Grade Level
Advanced College Level