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By a generalized inverse of a given matrix, we mean a matrix that exists fora larger class of matrices than the nonsingular matrices, that has some of theproperties of the usual inverse, and that agrees with inverse when given matrixhappens to be nonsingular. In theory, there are many different generalizedinverses that exist. We shall consider the Moore Penrose, weightedMoore-Penrose, Drazin and weighted Drazin inverses.  New determinantal representations of these generalized inverse based on theirlimit representations are introduced in this paper. Application of this newmethod allows us to obtain analogues classical adjoint matrix. Using theobtained analogues of the adjoint matrix, we get Cramer's rules for the leastsquares solution with the minimum norm and for the Drazin inverse solution ofsingular linear systems. Cramer's rules for the minimum norm least squaressolutions and the Drazin inverse solutions of the matrix equations ${\rm {\bfA}}{\rm {\bf X}} = {\rm {\bf D}}$, ${\rm {\bf X}}{\rm {\bf B}} = {\rm {\bf D}}$and ${\rm {\bf A}}{\rm {\bf X}}{\rm {\bf B}} ={\rm {\bf D}} $ are alsoobtained, where ${\rm {\bf A}}$, ${\rm {\bf B}}$ can be singular matrices ofappropriate size. Finally, we derive determinantal representations of solutionsof the differential matrix equations, ${\bf X}'+ {\bf A}{\bf X}={\bf B}$ and${\bf X}'+{\bf X}{\bf A}={\bf B}$, where the matrix ${\bf A}$ is singular.

Cramer's Rule for Generalized Inverse Solutions of Some Matrices Equations