My question is from Artin's algebra: Interpreting the matrix $\begin{bmatrix} 1 & 0\\ 0 & 2 \end{bmatrix}$ in $GL_2(\mathbb F_5)$, find the order of the orbit.
I'm not sure where to start, if I describe the orbit of this matrix normally, wouldn't it be all $2 \times 2$ matrices of the form $\begin{bmatrix} p & 0\\ 0 & q \end{bmatrix}$ with $p,q \in \mathbb R$? Would it be the same if the entries were interpreted in $\mathbb F_5$ as well?