TITLE: Minimal number of generators for $GL(n,\mathbb{Z})$
QUESTION [19 upvotes]: $\DeclareMathOperator{\gl}{GL}\DeclareMathOperator{\sl}{SL}$From de la Harpe's book "Topics in Geometric Group Theory" I learnt that $\gl(n,\mathbb{Z})$ is generated by the matrices $$s_1 = \begin{bmatrix} 0 & 0 & 0 & \cdots & 0 & 1 \\ 1 & 0 & 0 & \cdots & 0 & 0 \\ 0 & 1 & 0 & \cdots & 0 & 0 \\ \cdots & \cdots & \cdots & \cdots & \cdots & \cdots \\ 0 & 0 & 0 & \cdots & 0 & 0 \\ 0 & 0 & 0 & \cdots & 1 & 0 \end{bmatrix}$$ $$s_2 = \begin{bmatrix} 0 & 1 & 0 & \cdots & 0 & 0 \\ 1 & 0 & 0 & \cdots & 0 & 0 \\ 0 & 0 & 1 & \cdots & 0 & 0 \\ \cdots & \cdots & \cdots & \cdots & \cdots & \cdots \\ 0 & 0 & 0 & \cdots & 1 & 0 \\ 0 & 0 & 0 & \cdots & 0 & 1 \end{bmatrix}$$ $$s_3 = \begin{bmatrix} 1 & 1 & 0 & \cdots & 0 & 0 \\ 0 & 1 & 0 & \cdots & 0 & 0 \\ 0 & 0 & 1 & \cdots & 0 & 0 \\ \cdots & \cdots & \cdots & \cdots & \cdots & \cdots \\ 0 & 0 & 0 & \cdots & 1 & 0 \\ 0 & 0 & 0 & \cdots & 0 & 1 \end{bmatrix}$$ $$s_4 = \begin{bmatrix} -1 & 0 & 0 & \cdots & 0 & 0 \\ 0 & 1 & 0 & \cdots & 0 & 0 \\ 0 & 0 & 1 & \cdots & 0 & 0 \\ \cdots & \cdots & \cdots & \cdots & \cdots & \cdots \\ 0 & 0 & 0 & \cdots & 1 & 0 \\ 0 & 0 & 0 & \cdots & 0 & 1 \end{bmatrix}$$ whenever $n \geq 2$. It can be shown that $s_1, s_3, s_4$ suffice (a result by Hua and Reiner). The matrices $s_1$ and $s_3$ generate $\gl(n,\mathbb{Z})$ when $n$ is even and $\sl(n,\mathbb{Z})$ when $n$ is odd. But what about the odd case?

For odd $n$, is $\gl(n,\mathbb{Z})$ generated by two elements?

REPLY [7 votes]: Though it's a good exercise to work things out directly (as YCor does), the ideas here are all fairly old and can be extracted without much difficulty from the literature as Dietrich's reference indicates.    An old paper of Hua-Reiner, for instance, is freely available online here.   To supplement this with more details about generators, there is a discussion over more general rings than $\mathbb{Z}$ (but specialized frequently to $\mathbb{Z}$) in the first five sections of Chapter VII in the textbook by M. Newman, Integral Matrices, Academic Press, 1972.