I am working on the following problem.
Let $\mathbb{F}_q$ be a finite field with $q \neq 9$ elements and $a$ be a generator of the cyclic group $\mathbb{F}_q^{\times}$. Show that $\mathrm{SL}_2(\mathbb{F}_q)$ is generated by \begin{pmatrix} 1 & 1 \\ 0 & 1 \\ \end{pmatrix} and \begin{pmatrix} 1 & 0 \\ a & 1 \\ \end{pmatrix}
My attempt is to make upper-triangular matrixes and lower-triangular matrixes from these elements, because $\mathrm{SL}_2(K)$ is generated from triangular matrixes for any field $K$. When $q$ is prime, we can obtain every triangular matrixes by multiplying these two elements.
However, I don't see how to proceed when $q = p^n$. Since $|\mathrm{SL}_2(K)| = q^3-q$ and the order of two elements above is $p$, my attempt fails. I also don't see how the assumption that $q \neq 9$ works.