# Article

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Keywords:
commutativity; Drazin inverse; Moore-Penrose inverse; rank equality; matrix expression
Summary:
Necessary and sufficient conditions are presented for the commutativity equalities $A^*A^D = A^DA^*$, $A^{\dag }A^D = A^DA^{\dag }$, $A^{\dag }AA^D = A^DAA^{\dag }$, $AA^DA^* = A^*A^DA$ and so on to hold by using rank equalities of matrices. Some related topics are also examined.
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