Science:Math Exam Resources/Courses/MATH152/April 2010/Question A 15
[Category:MER Tag Eigenvalues and eigenvectors]
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Question A 15 

List all eigenvalues of . 
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 hints below. Read the first one and consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! If after a while you are still stuck, go for the next hint. 
Hint 1 

Eigenvalues are the zeros of characteristic polynomial where I is the identity matrix. 
Hint 2 

It is sometimes faster to use the fact that the sum and product of the eigenvalues equals the trace and determinant of the matrix, respectively. 
Checking a solution serves two purposes: helping you if, after having used all the hints, you still are stuck on the problem; or if you have solved the problem and would like to check your work.

Solution 1 

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Please rate my easiness! It's quick and helps everyone guide their studies. We calculate the characteristic polynomial. Setting we obtain Therefore the eigenvalues are and . 
Solution 2 

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Please rate my easiness! It's quick and helps everyone guide their studies. Using Hint 2 we can set up the following two equations: From here we quickly see the solutions and . 