ABSTRACT
We have already noted that only MP1 was needed when seeking solutions of a system of linear equations. This leads us to the idea of looking for “inverses” of A that satisfy only some of the MP-equations. We introduce some notation. Let A{1} = {G | AGA = A}, A{2} = {H | H AH = H}, and so forth. For example, A{1, 2} = A{1} ∩ A{2}. That is, a {1,2}-inverse of A is a matrix that satisfies MP1 and MP2. We have established previously that A{1, 2, 3, 4} has just one element in it, namely A+. Evidently we have the inclusions A{1, 2, 3, 4} ⊆ A{1, 2, 3} ⊆ A{1, 2} ⊆ A{1}. Of course, many other chains are also possible. You might try to discover them all. In this section, we devote our attention to A{1}, the set of all 1-inverses of A.
The idea of a 1-inverse can be found in the book by Baer [1952]. Baer’s idea was later developed by Sheffield [1958] in a paper. Let’s make it official.
