  We find a combinatorial formula which computes the first cotangent cohomology
module of Stanley-Reisner rings associated to matroids. For arbitrary
simplicial complexes we provide upper bounds for the dimensions of the
multigraded components of T^1. For specific degrees we prove that these bounds
are reached if and only if the simplicial complex is a matroid, obtaining thus
a new characterization for matroids. Furthermore, the graded first cotangent
cohomology turns out to be a complete invariant for nondiscrete matroids.
