Let me give some definitions first:
We define the inclusion $GL(n,\mathbb{Z})\to GL(n+1,\mathbb{Z})$ as $A\mapsto \begin{pmatrix}A&0\\0&1\end{pmatrix}$
We call $E(n,\mathbb{Z})$ the subgroup of $GL(n,\mathbb{Z})$ generated by elementary matrices (i.e. the ones with $1$ everywhere in the diagonal and a single entry $=1$ outside the diagonal)
Define the direct limits $GL(\mathbb{Z}):=\lim_\limits{\to}GL(n,\mathbb{Z})$ and $E(\mathbb{Z}):=\lim_\limits{\to}E(n,\mathbb{Z})$ with respect to the inclusion map written above.
It can be proved that $E(\mathbb{Z})$ is a normal subgroup of $GL(\mathbb{Z})$ and we call $K_1(\mathbb{Z}):=GL(\mathbb{Z})/E(\mathbb{Z})$.
My question is how can I show that the matrices $\begin{pmatrix}\pm 1&&&\\&\\&\pm1&&\\&&\ddots&\\&&&\pm1\end{pmatrix}$ cover all classes in $K_1(\mathbb{Z})$?
In simpler terms: how can I get all invertible matrices over $\mathbb{Z}$ by elementary row/column operation (up to enlarging the matrix and adding $1$'s in the right bottom corner)?
I know that $K_1(\mathbb{Z})$ is actually $\mathbb{Z}_2$, but I guess this particular fact can be shown directly and elementary.
Thank you!