Some Results for the Drazin Inverses of the Sum of Two Matrices and Some Block Matrices

Abstract

We give a formula of $(P+Q{)}^D$ under the conditions $P^{\mathrm{2}}Q+QPQ=\mathrm{0}$, $P^{\mathrm{3}}Q=\mathrm{0}$, and $PQPQ=\mathrm{0}$. Then, we apply it to give some expressions for the Drazin inverse of block matrix ($A$ and $D$ are square matrices) under some conditions, generalizing some recent results in the literature. Finally, numerical examples are given to illustrate our results.