Much of the traditional approach to linear model analysis is bound up in complex matrix expressions revolving about the usual generalized inverse. Motivated by this important role of the generalized inverse. the research summarized here began as an interest in understanding. in geometric terms. the four conditions defining the qnique Moore-Penrose Inverse. Such an investigation. it was hoped. might lead to a better understanding. and possibly a simplification of. the usual matrix expressions. Initially this research was begun by Francis Hsuan and Pat Langenberg, without knowledge of Kruskal's paper published in 1975. This oversight was perhaps fortu­ nate. since if they had read his paper they may not have continued their effort. A summary of this early research appears in Hsuan. Langenberg and Getson (1985). This monograph is a summary of the research on {2}-inverses continued by Al Getson. while a graduate student. in collaboration with Francis Hsuan of the Depart­ ment of Statistics. School of Business Administration. at Temple University. Philadelphia. The literature on generalized inverses and related topics is extensive and some of what is present here has appeared elsewhere. Generally. this literature is not presented from the point of view of {2}-inverses. We have tried to do justice to . the relevant published works and appologize for those we have either overlooked or possibly misrepresented.