# Similarity class of $3 \times 3$ matrices with entries in $\mathbb{F}_3$

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.

• Do you know the rational canonical form? Aug 16 '19 at 4:08
• yes, when I say analyze all possible cases I am referring to compute the different rational canonical forms once I determine all invariant factors.
– user437748
Aug 16 '19 at 4:11
• In $\Bbb F_3$ we have $x + 2 = x - 1$, so $x^3 - 1 = (x - 1)^3$. Aug 16 '19 at 4:45
• Ah, you are totally right!
– user437748
Aug 16 '19 at 11:49

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