In this paper, various properties of core-EP matrices are investigated. We introduce the MPDMP matrix associated with $A$ and by means of it, some properties and equivalent conditions of core-EP matrices can be obtained. Also, properties of MPD, DMP, and CMP inverses are studied and we prove that in the class of core-EP matrices, DMP, MPD, and Drazin inverses are the same. Moreover, DMP and MPD binary relation orders are introduced and the relationship between these orders and other binary relation orders are considered.

翻译：本文研究了核心-EP矩阵的多种性质。我们引入了与矩阵$A$相关的MPDMP矩阵，并借助该矩阵，得到了核心-EP矩阵的一些性质和等价条件。同时，研究了MPD、DMP和CMP逆的性质，并证明了在核心-EP矩阵类中，DMP逆、MPD逆与Drazin逆是相同的。此外，引入了DMP与MPD二元关系序，并探讨了这些序与其他二元关系序之间的联系。