Dense 2-generator subsemigroups of 2×2 matrices

We construct explicit examples of 2-generator dense subsemigroups of GL(2,R) and SL(2,R). In the complex case, we prove the existence of uncountably many 2-generator dense subsemigroups of GL(2,C). We also show that the semigroup of real linear fractional transformations on a proper subinterval of the real line does not admit any 2-generator dense subsemigroups, and then we construct a 3-parameter family of examples of 3-generator dense subsemigroups.