Science:Math Exam Resources/Courses/MATH221/December 2008/Question 07 (b)
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Question 07 (b) 

True or false (explain your answer): If A and B are similar matrices, then A and B have the same eigenvalues. 
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 two matrices are similar, then there is a relation between their respective characteristic polynomials (what is it?). How does the characteristic polynomial of a matrix relate to its eigenvalues? 
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Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. The statement is true: If A and B are similar, then there is an invertible matrix P for which . Then we can write
Therefore we compute that . This equality means precisely that the characteristic polynomials for A and B are identical. Since the eigenvalues of a matrix are the zeros of its characteristic polynomial, and the characteristic polynomials for A and B are identical, we conclude that the eigenvalues of A and B are the same. 