  We prove Poincar\'e and Sobolev inequalities in matrix A${}_p$ weighted
spaces. We then use these Poincar\'e inequalities to prove existence and
regularity results for degenerate systems of elliptic equations whose
degeneracy is governed by a matrix A${}_p$ weight. Such results parallel
earlier results by Fabes, Kenig, and Serapioni for a single degenerate equation
governed by a scalar A${}_p$ weight. In addition, we prove Cacciopoli and
reverse H\"older inequalities for weak solutions of the degenerate systems. As
a means to prove the Poincar\'e inequalities we prove that the Riesz potential
and fractional maximal function operators are bounded on matrix weighted $L^p$
spaces and go on to develop an entire matrix A${}_{p, q}$ theory.
