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The main result of this paper is the decidability of the membership problemfor $2\times 2$ nonsingular integer matrices. Namely, we will construct thefirst algorithm that for any nonsingular $2\times 2$ integer matrices$M_1,\dots,M_n$ and $M$ decides whether $M$ belongs to the semigroup generatedby $\{M_1,\dots,M_n\}$.  Our algorithm relies on a translation of the numerical problem on matricesinto combinatorial problems on words. It also makes use of some algebraicalproperties of well-known subgroups of $\mathrm{GL}(2,\mathbb{Z})$ and variousnew techniques and constructions that help to limit an infinite number ofpossibilities by reducing them to the membership problem for regular languages.

Decidability of the Membership Problem for $2\times 2$ integer matrices