The notion of the Moore–Penrose inverse of tensors with the Einstein product was introduced, very recently. In this paper, we further elaborate on this theory by producing a few characterizations of different generalized inverses of tensors. A new method to compute the Moore–Penrose inverse of tensors is proposed. Reverse order laws for several generalized inverses of tensors are also presented. In addition to these, we discuss general solutions of multilinear systems of tensors using such theory.

Reverse-order law for the Moore–Penrose inverses of tensors is useful in the field of multilinear algebra. In this paper, we first prove some more identities involving the Moore–Penrose inverses of tensors. We then obtain a few necessary and sufficient conditions of the reverse-order law for the Moore–Penrose inverses of tensors via the Einstein product.