The least squares g-inverse for the sum of multiple matrices

In this paper, we give the necessary and sufficient conditions for the set identities (A1+A2+⋯+Ak){1,3}=A1{1,3}+A2{1,3}+⋯+Ak{1,3} and (A1+A2+⋯+Ak){1,4}=A1{1,4}+A2{1,4}+⋯+Ak{1,4} which concern the least squares g-inverse ({1,3}-inverse) and the minimum norm g-inverse ({1,4}-inverse) of sum of k matrices.