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Find all $n\in\mathbb{N}$ such that there exists $\mbox{2x2}$ integer matrix $A$ (not being an identity matrix) such that $A^n=I_2$.

Any hint please?

luka5z
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    This is a duplicate of http://math.stackexchange.com/questions/534198/a-2-times-2-matrix-a-such-that-an-is-the-identity-matrix?rq=1 – Terra Hyde Oct 27 '15 at 17:31
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    Try thinking of what matrix multiplication does to the plane, and remembering that matrix multiplication is function composition. So if a matrix $A$ rotates the plane by $90^\circ$, $A^4$ rotates by $360^\circ$, which is the identity. – 211792 Oct 27 '15 at 17:32
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    @Terra it's not a duplicate... – luka5z Oct 27 '15 at 17:33
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    I see that now. You want only integers. That may change the answer substantially. Apologies. – Terra Hyde Oct 27 '15 at 17:37
  • @Terra, no worries – luka5z Oct 27 '15 at 17:39
  • Why is anyone asking you this? Look up the modular group. You are also considering determinants $-1,$ not sure how much worse that makes it. You can get $n=2,3,4,6,$ do not currently think you can get 12. – Will Jagy Oct 27 '15 at 18:56
  • I believe all multiplies of $4$ work. Take rotation matrix (90 deg) – luka5z Oct 27 '15 at 19:03
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    Yes, but that actually satisfies $A^4 = I.$ If you are not going to use the smallest $n$ that works, then you can get $2n,3n,4n,6n.$ I see, the only restriction is $A \neq I.$ Again, where did you get the problem? – Will Jagy Oct 27 '15 at 19:06
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    It's a test task for Phd in mathematics candidates – luka5z Oct 27 '15 at 19:15

1 Answers1

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Hints.

  1. If $A^n=I$, what are the possible eigenvalues of $A$ over $\mathbb C$?
  2. In view of (1), since the trace of $A$ must be an integer, and any non real eigenvalues of $A$ must occur in conjugate pairs, can you narrow down the possible eigenvalues of $A$?
  3. The candidate eigenvalues in (2) are realisable by integer matrices. Why? (Think companion matrix.) Now, what can you say about $n$?
user1551
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    @luka5z Yes. So, there are two possibilities: 2(a) the eigenvalues belong to the set ${-1,1}$, or 2(b) the spectrum must be in the form of $\cos\theta\pm i\sin\theta$ such that $\sin\theta\ne0$. In the latter case, since $2\cos\theta$ (the trace of $A$) has to be an integer, you can say something more... – user1551 Oct 28 '15 at 13:36