Find a finite generating set for $Gl(n,\mathbb{Z})$ I need to find a finite generating set for $Gl(n,\mathbb{Z})$. I heard somewhere once that this group is generated by the elementary matrices - of course, if I'm going to prove that $GL(n,\mathbb{Z})$ has a finite generating set, I would need to prove that any matrix $M\in  GL(n,\mathbb{Z})$ can be generated by only finitely many of them.
At first, I didn't have a clue as to how to do this, so I did a bit of scouring the internet for any information that might be useful. There were a few proofs or hints at proofs, including here on MSE and also on MathOverflow, but they were either too advanced, didn't give enough details, assumed theory I can't assume at this point (for example about rings or principle ideal domains), or were extremely complicated (as in 4 pages with 4 lemmas that needed to be proven first - and this example didn't even prove exactly what the finite generator of $GL(n,\mathbb{Z})$ is).
This looks promising. In their notation, essentially, if $n$ is even, then $GL(n,\mathbb{Z})$ is generated by $s_{1}$ and $s_{3}$ And when $n$ is odd, $-s_{1}$ and $s_{3}$ generate $GL(n,\mathbb{Z})$, where $s_{1}=\begin{pmatrix} 0&0&0&\cdots &0&1\\ 1&0&0&\cdots & 0&0\\0&1&0&\cdots & 0 &0 \\ \vdots & \vdots & \vdots & & \vdots &\vdots \\ 0&0&0&\cdots & 0&0\\ 0&0&0&\cdots &1 & 0\end{pmatrix}$ and $s_{3}=\begin{pmatrix} 1&1&0&\cdots &0&0\\ 0&1&0&\cdots & 0&0\\0&0&1&\cdots & 0 &0 \\ \vdots & \vdots & \vdots & & \vdots &\vdots \\ 0&0&0&\cdots & 1&0\\ 0&0&0&\cdots &0& 1\end{pmatrix}$
How is a relatively simple way to prove this, that does not invoke rings or ideals at all (only group theory is permissible), and does not amake reference so to papers or $Hom(G,C_{p})$ (whatever that is)?
I'm guessing since the group operation in $GL(n,\mathbb{Z})$ is matrix multiplication, I'm guessing I'd have to show that any matrix $A$ can be generated by multiplying various combinations of $s_{1}$ and $s_{3}$ in the case when $n$ is even and various combinations of $-s_{1}$ and $s_{3}$ in the case when $n$ is odd. But what do those combinations look like when we're dealing with matrix multiplication? Do they include scalar multiples like integer linear combinations when the operation is addition? And how do we know what order to put them in, since matrix multiplication is not commutative?
Thank you.

The elementary matrices generate $GL(n, \mathbb{Z})$. Use the row-switching transformations to reduce to the case $n = 2$. Then $SL(2, \mathbb{Z})$ is a finite-index subgroup of $GL(2, \mathbb{Z})$, and it's an amalgamated free product of two finite groups (see Serre's Trees, for example).
