The main task of the project is the development of methods of quaternion algebras  in the study of quaternion generalized inverse matrices and their applications in solving quaternion matrix equations with  different restrictions  of   solution spaces. The Moore-Penrose (MP-)inverse, the (D-)Drazin inverse, and the core (C-)inverse are most well-known generalized inverses. Their compositions induce new generalized inverses, in particular DMP, MPD, CMP,  MPCEP  inverses, and others that have their  special characterizations, representations, and applications in particular as tools in solving matrix equations. One of direct methods of constructions of generalized inverses  are their determinantal representations. In the case of quaternion matrices, the problem of defining a determinant with noncommutative entries arises. The project uses the theory of column and row noncommutative determinants (recently developed by the author) that are generalizations of the Moore determinant, defined only for Hermitian matrices. Using row-columnes determinants, the project will develop the theory of quaternion generalized inverse matrices, in particular, various compositions of weighted core-EP, MP-, and D- inverses. In consequence of obtained determinatal representations of quaternion generalized inverses, solutions of related restricted quaternion matrix equations and minimization matrix problems will be expressed by analogs of Cramer's rule. The relevance of the project can be confirmed   that   quaternion matrices and generalized inverses are the objects of active researches and applications in a wide range of  areas nowadays.