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I just got a quick practice question here that I think should be simple but I can't find a definitive answer.

Let $A$ be a square matrix such that $A^3=A$. What can you say about the eigen values of $A$?

It is multiple choice and all of the answers are combinations of -1, 1, and 0. I'm pretty sure 0 and 1 are possible but I'm not sure how to prove any of them. Thanks in advance.

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Hint: Write $A^3 - A = A(A^2 - I) = A(A+I)(A-I) = 0$. – Amzoti Mar 26 '13 at 1:44
Let $v$ be an eigenvector with eigenvalue $c$. Then $cv=Av=(A^3)v=(c^3)v$. So what can you say about $c$? – Relsiark Mar 26 '13 at 1:46
Oh ok, thanks to both of you for the help. It makes it pretty straight forward. – Chance Mar 26 '13 at 1:53
In general, the eigenvalues are roots of the minimal polynomial for $A$. If $A^3=A$, what do we know about the minimal polynomial for $A$? – Thomas Andrews Mar 26 '13 at 2:09
Chance, if you can solve the problem now, write up an answer and post it. Then after a while you can accept your answer. This is encouraged on this website. – Gerry Myerson Mar 26 '13 at 12:17
up vote 2 down vote accepted

Minimal polynomial of matrix $A$ satisfying $A^3=A$ is given by $x^3-x=0$,so the possible eigenvalues will be $0,1,-1$, depending on the given matrix now, we can select it from MCQ.

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It's not necessarily the minimal polynomial; what you can say is that the minimal polynomial is a factor of $x^3-x$. However, the conclusion about the possible eigenvalues is correct; one can't say more because it's not possible to say what the minimal polynomial or the characteristic polynomial are. – egreg May 22 '13 at 17:34

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