Reverse order laws for the generalized inverses of multiple matrix products

AbstractBy using the ranks of matrices, this article gives necessary and sufficient conditions for the reverse product A+n…A+2A+1of the Moore-Penrose inverses of A1, A2, …, An to a {1}-inverse (inner inverse), {1, 2}-inverse (reflexive inner inverse), {1, 3}-inverse, {1, 4}- inverse, {1, 2, 3}-inverse, {1, 2, 4}-inverse, or Moore-Penrose inverse of matrix product A=A1A2…An