Science:Math Exam Resources/Courses/MATH221/December 2008/Question 02 (b)
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Question 02 (b) 

True or false (explain your answer): if is an matrix such that is an eigenvalue of , then is not invertible. 
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Hint 

Science:Math Exam Resources/Courses/MATH221/December 2008/Question 02 (b)/Hint 1 
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Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. This is true. is not invertible when . We know that the characteristic polynomial is . If , then . Therefore, only if . However, we know that because the eigenvalue, is the root of the polynomial equation. . Therefore, if , and this shows that if , then is not invertible 