The first cotangent cohomology module $T^1$describes the first order deformations of a commutative ring. The second cotangent cohomology module contains the obstructions to lifting such deformations. For Stanley-Reisner rings, these modules have an explicit description: their multigraded components are given by the relative cohomology of some topological spaces associated to the defining simplicial complex. In this talk I will focus on Stanley-Reisner rings associated to matroids. Among other results, I will show that $T^1$ provides a new complete characterization for matroids.