  We consistently develop a recently proposed scheme of matrix extension of
dispersionless integrable systems for the general case of multidimensional
hierarchies, concentrating on the case of dimension $d\geqslant 4$. We present
extended Lax pairs, Lax-Sato equations, matrix equations on the background of
vector fields and the dressing scheme. Reductions, construction of solutions
and connections to geometry are discussed. We consider separately a case of
Abelian extension, for which the Riemann-Hilbert equations of the dressing
scheme are explicitly solvable and give an analogue of Penrose formula in the
curved space.
