  We study degenerate Sobolev spaces where the degeneracy is controlled by a
matrix $A_p$ weight. This class of weights was introduced by Nazarov, Treil and
Volberg, and degenerate Sobolev spaces with matrix weights have been considered
by several authors for their applications to PDEs. We prove that the classical
Meyers-Serrin theorem, H = W, holds in this setting. As applications we prove
partial regularity results for weak solutions of degenerate p-Laplacian
equations, and in particular for mappings of finite distortion.
