As a central topic certain relations between operator matrices are investigated which are called delta relations. The main aim of these relations is to reduce questions about classes of operators without invertibility symbol to those which admit an invertibility symbol. Particular attention is devoted to the generalized inversion of such operators. Different kinds of relations are introduced in order to analyze the "information" contained in the symbols of the related operators. Several examples are considered and the theory is also applied to singular integral operators with Carleman shift. Asymptotic solutions of equations characterized by those operators are presented. The approach simplifies several known results, makes the theory more rigorous from the operator theoretic point of view, and allows further conclusions in a very compact form.