The reverse order law for the generalized inverse over a skew field

Abstract

Abstract In this paper, we study the reverse order law for the generalized inverse of the product of three matrices over a skew field with ranks of those matrices discussed, and give a necessary and sufficient condition under which the reverse order law holds. From this, we obtain necessary and sufficient conditions for the reverse order laws for the known generalized inverses, such as the Moore–Penrose inverse, the Drazin inverse and the group inverse, over skew fields.