Question: Let $SL_2(\mathbb{Z}/3\mathbb{Z})$ be $SL_2(\mathbb{Z})$ with the matrix entries interpreted modulo 3. It is a group. What is the order of $\begin{bmatrix}1&1\\0&1\end{bmatrix}$ in $SL_2(\mathbb{Z}/3\mathbb{Z})$?
I'm not entirely sure how to proceed with this. To my understanding, normally you would compose the matrix with itself until you get the matrix itself back as a result. But i'm not entirely sure what exactly would be different about $\mathbb{Z}/3\mathbb{Z}$..
Actually, the order of $\left[\begin{smallmatrix}1&1\\0&1\end{smallmatrix}\right]$ is the smallest $n\in\mathbb N$ such that $\left[\begin{smallmatrix}1&1\\0&1\end{smallmatrix}\right]^n=\left[\begin{smallmatrix}1&0\\0&1\end{smallmatrix}\right]$.
You have$$(\forall n\in\mathbb N):\begin{bmatrix}1&1\\0&1\end{bmatrix}^n=\begin{bmatrix}1&n\\0&1\end{bmatrix}$$and therefore the answer is $3$, since, in $\mathbb Z_3$,$$\begin{bmatrix}1&3\\0&1\end{bmatrix}=\begin{bmatrix}1&0\\0&1\end{bmatrix}$$(but $\left[\begin{smallmatrix}1&2\\0&1\end{smallmatrix}\right]\neq\left[\begin{smallmatrix}1&0\\0&1\end{smallmatrix}\right]$).