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The main result of this paper is the decidability of the membership problem for 2x2 nonsingular integer matrices. Namely, we will construct the first algorithm that for any nonsingular 2x2 integer matrices M1,...,Mn and M decides whether M belongs to the semigroup generated by {M1,...,Mn}. Our algorithm relies on a translation of numerical problems on matrices into combinatorial problems on words. It also makes use of some algebraic properties of well-known subgroups of GL(2,Z) and various new techniques and constructions that help to convert matrix equations into the emptiness problem for intersection of regular languages

Decidability of the Membership Problem for 2 × 2 integer matrices