Science:Math Exam Resources/Courses/MATH152/April 2022/Question A20
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Question A20 

Let be a matrix with eigenvalues , . Find . (Hint: how does transform the eigenvectors?) 
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 

Looking at how acts on the eigenvectors, you should be seeing something that looks like , , . Using this, and the fact that linearly independent eigenvectors span , what can you conclude about ? 
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Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. We start by looking at how transforms the eigenvectors. We don’t need to know what the eigenvectors are to do this, we just need the relation .
Because and are eigenvectors associated with distinct eigenvalues, we know they are linearly independent, and span . And, we have that transforms and in the same way. So we must have that is the diagonal matrix given by 