Cramer’s Rules for the System of Two-Sided Matrix Equations and of Its Special Cases

Within the framework of the theory of row-column determinants previously introduced by the author, we get determinantal representations (analogs of Cramer’s rule) of a partial solution to the system of two-sided quaternion matrix equations A1XB1=C1, A2XB2=C2. We also give Cramer’s rules for its special cases when the first equation be one-sided. Namely, we consider the two systems with the first equation A1X=C1 and XB1=C1, respectively, and with an unchanging second equation. Cramer’s rules for special cases when two equations are one-sided, namely the system of the equations A1X=C1, XB2=C2, and the system of the equations A1X=C1, A2X=C2 are studied as well. Since the Moore-Penrose inverse is a necessary tool to solve matrix equations, we use its determinantal representations previously obtained by the author in terms of row-column determinants as well

Explicit Determinantal Representation Formulas of W

By using determinantal representations of the W-weighted Drazin inverse previously obtained by the author within the framework of the theory of the column-row determinants, we get explicit formulas for determinantal representations of the W-weighted Drazin inverse solutions (analogs of Cramer’s rule) of the quaternion matrix equations WAWX=D, XWBW=D, and W1AW1XW2BW2=D

Determinantal Representations of the Core Inverse and Its Generalizations

Generalized inverse matrices are important objects in matrix theory. In particular, they are useful tools in solving matrix equations. The most famous generalized inverses are the Moore-Penrose inverse and the Drazin inverse. Recently, it was introduced new generalized inverse matrix, namely the core inverse, which was late extended to the core-EP inverse, the BT, DMP, and CMP inverses. In contrast to the inverse matrix that has a definitely determinantal representation in terms of cofactors, even for basic generalized inverses, there exist different determinantal representations as a result of the search of their more applicable explicit expressions. In this chapter, we give new and exclusive determinantal representations of the core inverse and its generalizations by using determinantal representations of the Moore-Penrose and Drazin inverses previously obtained by the author