I am doing an exercise where I have to use that $(GL(2,\mathbb{Z}\_3),\cdot)$ is group. I just have a hard time accepting that this a group. $GL(2,\mathbb{Z}\_3)$ is described as "the set of invertible $2 \times2 $ matrices with coefficients in $\mathbb{Z}\_3$". The $\cdot$ is simply the ordinary matrix multiplication operator. I don't understand how the group axiom that every element in $(GL(2,\mathbb{Z}\_3),\cdot)$ has an inverse in $(GL(2,\mathbb{Z}\_3),\cdot)$ is satisfied. For example the matrix
$$\begin{bmatrix} 2 & 1 \\ 1 & 2 \end{bmatrix} \in GL(2,\mathbb{Z}\_3)$$ has inverse
$$\begin{bmatrix} 2/3 & -1/3 \\ -1/3 & 2/3 \end{bmatrix}.$$
Now this matrix should be in $(GL(2,\mathbb{Z}\_3)$?

Why?

I guess since we are working with coefficients in $\mathbb{Z}\_3$, we should take modulo 3 on all the coefficients, but the problem here is that that's not possible since $gcd(3,3)=3 \neq 1$.

I suspect that I have some fundamental misunderstanding; can anyone explain what I am missing here?

Thanks