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I've been trying to solve the following problem.

Find a representative for each similarity class of $3 \times 3$ matrices $A$ with entries in $\mathbb{F}_3 = \mathbb{Z}/3\mathbb{Z}$ such that $A^4 = A$.

My idea: if $A$ is an invertible matrix, then $A^3 = I$ which implies that the minimum polynomial of $A$ divides $x^3 - 1 = (x - 1)(x + 2)(x + 2)$. At this point, I can analyze all the possible cases and determine from these cases the similarity classes. However, if $A$ is not invertible then I will have too many cases to analyze; so I am not sure if I am approaching this from the wrong perspective.

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    $\begingroup$ Do you know the rational canonical form? $\endgroup$ Aug 16 '19 at 4:08
  • $\begingroup$ yes, when I say analyze all possible cases I am referring to compute the different rational canonical forms once I determine all invariant factors. $\endgroup$
    – user437748
    Aug 16 '19 at 4:11
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    $\begingroup$ In $\Bbb F_3$ we have $x + 2 = x - 1$, so $x^3 - 1 = (x - 1)^3$. $\endgroup$ Aug 16 '19 at 4:45
  • $\begingroup$ Ah, you are totally right! $\endgroup$
    – user437748
    Aug 16 '19 at 11:49
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You don't need to consider the invertible case separately. The minimal polynomial of $A$ must divide $x^4-x=x(x^3-1)=x(x-1)^3$ over $\mathbb F_3$. This will give the possible forms for the characteristic polynomial, which has degree $3$ and must have exactly the same irreducible factors as the minimal polynomial. There are many cases, but it's not that complicated.

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    $\begingroup$ +1, but I only get $7$ cases, is that really "many"? $\endgroup$ Aug 16 '19 at 15:43

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