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.
Best Answer
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.