Science:Math Exam Resources/Courses/MATH221/April 2009/Question 12 (e)
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Question 12 (e) 

The sum of two eigenvectors of A is again an eigenvector of A. 
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Hint 

Science:Math Exam Resources/Courses/MATH221/April 2009/Question 12 (e)/Hint 1 
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Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. The answer is false. The motivation for why it should be false is as follows: Let be two eigenvectors of A with respective eigenvalues . Then . So unless , will not be an eigenvector. The above observation will guide us in our construction of a counterexample. Consider
The eigenvalues of A are 1 and 2 with corresponding eigenvectors and . However, as discussed above, . So is not an eigenvector of A. 