Some Formulas on the Drazin Inverse for the Sum of Two Matrices and Block Matrices

In this article, we construct new explicit formulas for the Drazin inverse of P+Q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P+Q$$\end{document} under some conditions, where P and Q are singular square matrices over skew fields. These formulas generalize some recent results in literature. By using these formulas, we establish new results for the Drazin inverse of 2×2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2\times 2$$\end{document} block matrix under some assumptions over skew fields. Finally, illustrative numerical examples over skew fields are presented to demonstrate results of this paper.