Three things: The IC p[0] == 0 makes p[t] a scalar; should it be a matrix? The operator * does elementwise multiplication; you probably want matrix multiplication . (Dot). You probably need an explicit matrix for H, too.
Finally, I'm not sure DSolve is even programmed to solve a properly coded matrix ODE in terms of a matrix-valued variable p. NDSolve can solve them numerically, provided the parameters H and r are explicit numerical matrices/scalars. For DSolve, you might also try an explicit symbolic matrix for p[t] like {{p11[t], p12[t]}, {p21[t], p22[t]}}.
