In this article, we present some formulas of the Drazin inverses of the sum of two matrices under the conditions P 2 QP = 0, P 2 Q 2 = 0, QPQ = 0 and PQP 2 = 0, PQ 2 = 0, QP 3 = 0 respectively. These conditions are weaker than those used in some literature on this subject. Furthermore, we apply our results to give the representations for the Drazin inverses of block matrix A B C D (A and D are square matrices) with generalized Schur complement is zero.

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Abstract. The relations between these classes and some basic properties of these operators are studied in this study.Mathematics subject classification (2010): 47B15, 47B20, 47A15, 15A09.

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