Science:Math Exam Resources/Courses/MATH221/December 2008/Question 05 (a)
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Question 05 (a) 

Find the eigenvalues and eigenvectors for

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 

Recall that if , then is an eigenvalue of with eigenvector . 
Hint 2 

First, calculate the eigenvalues and . To find the corresponding eigenvectors, find vectors in the kernel of and . 
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 

Found a typo? Is this solution unclear? Let us know here.
Please rate my easiness! It's quick and helps everyone guide their studies. To calculate the eigenvalues of we need to find values of such that has a nontrivial kernel. A good way to check this is to look for zeros of the determinant of the matrix above:
Luckily for us, the polynomial is already factorized, so that we can simply read off the eigenvalue and . To find the eigenvector corresponding to the eigenvalue , we solve . This equation is
and thus is an eigenvector associated to . Repeating the same steps for gives us the eigenvector equation
and thus is an eigenvector corresponding to . To summarize, is an eigenvalue of with corresponding eigenvector , and is an eigenvalue of with corresponding eigenvector . 