On a family of matrix equalities that involve multiple products of generalized inverses

One of the fundamental matrix equalities that involve multiple products of matrices and their generalized inverses is given by A(1)B(1)(-) A(2)B(2)(-) . . .A(k)B(k)(-) A(k) (+1) = A, where A(1), B-1, A(2), B-2, ..., A(k), B-k, A(k+1), and Aare matrices of appropriate sizes, and B-1(-) ,B-2(-) ,.. ., B-k(-) are generalized inverses of matrices. Many matrix equality problems related to generalized inverses, including various reverse-order laws for generalized inverses of matrix products, can be reduced to the special situations of this general equality. Generally speaking, this equality does not necessarily hold, and therefore, it is a primary task to establish necessary and sufficient conditions for it to hold for some generalized inverses of matrices, or to always hold for all generalized inverses of matrices. The two basic cases of the equality problem for k = 1,2 and their special forms were well known and approached in the discipline of generalized inverses of matrices. As a further exploration on this kind of matrix, this note explores performances of the equality for k = 3. We shall present an algebraic procedure to derive explicit necessary and sufficient conditions for the equality to always hold through the skilful use of rank equalities for some block matrices that are constructed from the given matrices, and then mentions the ideas and techniques of block matrix representations in the characterization of the above general matrix equality.