$[Solved]$ Find all units in the ring $M_2(\mathbb{Z})$

Question

my problem here is:

Find all units in the ring $M_2(\mathbb{Z})$

I've done it before but now it's a matrix and I'm lost. First I think on the determinant because I saw some other wrote about it, but I'm not sure. Any help?

Thank you

Answer 1

More generally, for any commutative ring $R$, an $n\times n$ matrix $M$ lies in $\mathbf{GL}(n,R)$ if and only if $\det M\in R^{\times}$ (the units of $R$).

The units of $\mathbf Z$ are indeed only $1$ and $-1)$, simply because if $|u|>1$, you cannot find another integer $v$ such that $uv=1$ since $|uv|=|u||v|>|v|$.