Science:Math Exam Resources/Courses/MATH152/April 2022/Question A14
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Question A14 

Find a matrix that has eigenvectors and , corresponding to eigenvalues and . 
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 

We know that for a matrix , the first column is , and the second column is . How can you relate the standard basis vectors and to the given eigenvectors? 
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. Recall from the hint that a matrix will have first column , and second column , where and are the standard basis vectors and . Now, note that , and use the property that for an eigenvalueeigenvector pair to get since . This results in the first column of simply being . For the second column of , we first note that . Using the same eigenvalueeigenvector relation we have since . This results in the second column being , and the matrix is 