Abstract: In this paper, we prove a decomposition theorem of matrix pair over an arbitrary skew field. By applying the decomposition theorem, we obtain some neccessary and sufficient conditions of reverse order laws for g-inverse and reflexive g-inverse of matrix products on an arbitrary skew field.

Key words: matrix decomposition, g-inverse, reflexive g-inverse, reverse order law, Skew field, matrix decomposition, g-inverse, reflexive g-inverse, reverse order law