Knowing that for $n \geq 2$, $\mathrm{GL}(n, \mathbb{Z}) = \big\{ A \in \mathrm{M}_{n,n}(\mathbb{Z}) \mid \det(A) \in \{ 1, −1 \} \big\}$ is a group with respect to matrix multiplication, prove that for every integer $n \geq 2$ the group $\mathrm{GL}(n, \mathbb{Z})$ is finitely generated.
If I prove that $\mathrm{GL}(n, \mathbb{Z})$ has finite subgroups does that mean it has a finite set of generators so that it is finitely generated?