Science:Math Exam Resources/Courses/MATH152/April 2015/Question A 23
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Question A 23 |
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The matrix has the eigenvalue . Find all eigenvectors corresponding to this 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 |
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The eigenvectors of eigenvalue satisfy the equation ; in other words, Find all solutions of this linear system. |
Checking a solution serves two purposes: helping you if, after having used the hint, you still are stuck on the problem; or if you have solved the problem and would like to check your work.
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Solution |
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Please rate my easiness! It's quick and helps everyone guide their studies. We would like to find the eigenvectors of the matrix corresponding to the eigenvalue . Recall (from the hint or otherwise) that the eigenvectors of corresponding to the eigenvalue are those vectors such that . When , this equation reads We solve this system using row reduction. Beginning with the augmented system matrix we subtract half the first row from the second and third rows, and then divide the first row by 10, giving Therefore, any vector satisfying ( is free) will be an eigenvector, i.e., the desired eigenvectors are |