© 2017 Elsevier Inc. This paper derives some further results on the core inverse of matrices and the core partial ordering. First we treat two representations of core inverse obtained by Baksalary and Trenkler (2010) and derive maximal classes of matrices for which the representations are valid. Next we discuss several formulas on the core partial ordering presented by Malik (2013), and extend them by deriving maximal classes of matrices for which the formulas remain true.