Invariance property of a five matrix product involving two generalized inverses

Matrix expressions composed by generalized inverses can generally be written as f(A⁻₁, A⁻₂, . . ., A⁻_k), where A₁, A₂, . . ., A_k are a family of given matrices of appropriate sizes, and (·)⁻ denotes a generalized inverse of matrix. Once such an expression is given, people are primarily interested in its uniqueness (invariance property) with respect to the choice of the generalized inverses. As such an example, this article describes a general method for deriving necessary and sufficient conditions for the matrix equality A₁A⁻₂A₃A⁻₄A₅ = A to always hold for all generalized inverses A⁻₂ and A⁻₄ of A₂ and A₄ through use of the block matrix representation method and the matrix rank method, and discusses some special cases of the equality for different choices of the five matrices.