Science:Math Exam Resources/Courses/MATH152/April 2022/Question A15
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Question A15 

For which value of does the matrix below have two linearly independent eigenvectors corresponding to the eigenvalue ? 
Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you? 
If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! 
Hint 

If has 2 linearly independent eigenvectors corresponding to , then the nullspace of must be 2dimensional; equivalently, we must have . Is there a simple way to pick so that ? 
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Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. Since we want to have 2 linearly independent eigenvectors associated with , then we would have . This means that using row operations, we should be able to reduce the matrix to a single row! We need to find such that this holds. Then, in order to have reduce to a single row, we take . 