An explicit decomposition formula of a matrix in $G{L}_2\left(\mathbb{Z}\right)$

Abstract

Given $A:=\left(\begin{array}{cc}1& 1\\ 0& 1\end{array}\right)$, $B:=\left(\begin{array}{cc}1& 0\\ 1& 1\end{array}\right)$ and $C:=\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right)$ three elements of $G{L}_2\left(\mathbb{Z}\right)$, we propose an explicit formula that provides the decomposition of any $M\in G{L}_2\left(\mathbb{Z}\right)$ in $⟨A,B,C⟩$.