Science:Math Exam Resources/Courses/MATH307/December 2005/Question 04 (b)
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Question 04 (b) 

Let A and B be any two matrices. Show that if is an eigenvalue of AB and then is also an eigenvalue of BA. 
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 

Let v be the associated eigenvector of AB. Use the vector . 
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.

Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. Let v be an eigenvector of AB with eigenvalue λ, ABv = λv. Using the vector , we see that
Hence by definition, is an eigenvalue for (with eigenvector ). (Alternatively, one can view the above proof as
and then again we see that is an eigenvalue for matrix associated to the vector . 