Use the rational canonical form to determine all conjugacy classes of $GL_2(\mathbb{F}_3)$
I know for a given conjugacy class in $GL_2(\mathbb{F}_3)$, pick any representative $A$, we have $$\mathbb{F}_{3 A}^2\simeq \mathbb{F}_3[x]/(f_1)\oplus\mathbb{F}_3[x]/(f_2)\oplus\dots\oplus \mathbb{F}_3[x]/(f_n).$$ Such form is unique for each conjugacy class. But I don't see how I can find all conjugacy classes based on this invariant factor decomposition