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I am studying for a linear algebra exam and have come across a question that states:

For each $n \ge 1$ find the number of similarity classes of 2 x 2 matrices with integer entries of order exactly $n$ - ie matrices such that $A^n = I$ but $A^k \not = I$ for any $k < n$ with integer entries.

By similarity classes, I think the question means conjugacy classes since matrices are similar if there exists $P$ such that $B = PAP^{-1}$

I am completely stuck on how to start this problem. Any help would be appreciated!

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Depending on which level you're on there's different approaches.

If you know complex matrices and the matrices are supposed to be complex matices

Then you can observe that the eigenvalues for the matrix has to satisfy the same equation, that is $\lambda^k = 1$ first when $k=n$.

If it's real matrices

You have to show that a matrix $A$ is either:

  1. a (possibly non-uniform) scaling. That is that $Ax = \lambda_u u\cdot x u + \lambda_v v\cdot x v$ for some (eigen)vectors $u$ and $v$ and (eigen)values $\lambda_u$ and $\lambda_v$
  2. a scaled rotation. That is $A = cR$ for some rotation matrix $R$

After that you could show that case one is only possible if $A = I$ (for $n=1$) and $A=-I$ (for $n=2$), but that would make it fit the second type anyway.

The second type is only possible if $c=1$ and $R$ is a rotation of angle $2r\pi/n$, (the case $c=-1$ is also possible, but that would be covered by modifying the rotation by the angle $\pi$).

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