Science:Math Exam Resources/Courses/MATH307/December 2005/Question 03 (a)
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Question 03 (a) 

Decide whether the following statement is true or false. You need not give a reason. All matrices in this question are square . If then every eigenvalue of A is either 1 or 1. 
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 

Science:Math Exam Resources/Courses/MATH307/December 2005/Question 03 (a)/Hint 1 
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 1 

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Please rate my easiness! It's quick and helps everyone guide their studies. The answer is true. We are given that which implies that . Let is the characteristic polynomial of A and its minimal polynomial. Notice that divides (since the matrix itself satisfies its minimal polynomial by the CayleyHamilton theorem). Furthermore, p and m also share roots and these form the only roots. Hence all the roots of p must be a subset of the roots of which are . These roots are the eigenvalues of the matrix A and so the matrix can only have these eigenvalues. 
Solution 2 

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Please rate my easiness! It's quick and helps everyone guide their studies. This statement is true. If λ is an eigenvalue of A, then 1/λ is an eigenvalue of A^{1} with the same eigenvectors (Av = λv, v = λA^{1}v, v/λ = A^{1}v). Hence or, equivalently (since v ≠ 0), so λ = 1 or λ = 1. 