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Question A 24 

A matrix A with real entries has been typed into MATLAB. The result of the command = eig(A) is (after some slight formatting changes to make it fit better in the exam):
V = 0.8165 + 0.0000i 0.8165 + 0.0000i 0.5774 + 0.0000i 0.0000 + 0.0000i 0.0000 + 0.0000i 0.5774 + 0.0000i 0.4082  0.4082i 0.4082 + 0.4082i 0.5774 + 0.0000i D = 1.0000 + 2.0000i 0.0000 + 0.0000i 0.0000 + 0.0000i 0.0000 + 0.0000i 1.0000  2.0000i 0.0000 + 0.0000i 0.0000 + 0.0000i 0.0000 + 0.0000i 1.0000 + 0.0000i Circle all true statements below:

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 

Definition of eigenvectors and eigenvalues. In Matlab, command returns diagonal matrix of eigenvalues and matrix whose columns are the corresponding right eigenvectors, so that Note also that there are no repeated diagonal entries in . This can be used for the question. 
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Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. Following the hint, the diagonal matrix is the eigenvalues and the column of are the are the corresponding right eigenvectors. Therefore, A has 3 eigenvalues: , , , and their corresponding eigenvectors are , , . The eigenvectors in are normalized so that the 2norm of each one is .
is wrong. We have is correct. A has three different eigenvalues and hence the three eigenvectors are independent, and they can form a set of basis. is correct. is a scalar multiples of the last column vectors of . The scalar multiplication of any eigenvector is a eigenvector as well, so is correct Thus, the answer is . 