Analogues of the adjoint matrix for generalized inverses and corresponding Cramer rules

In this article, we introduce determinantal representations of the Moore - Penrose inverse and the Drazin inverse which are based on analogues of the classical adjoint matrix. Using the obtained analogues of the adjoint matrix, we get Cramer rules for the least squares solution and for the Drazin inverse solution of singular linear systems. Finally, determinantal expressions for ${\mathbf{A}}^{+}\mathbf{A}$, $\mathbf{A}{\mathbf{A}}^{+}$, and ${\rm {\bf A}}^{D} {\rm {\bf A}}$ are presented.