Necessary and sufficient conditions are presented for the commutativity equalities $A^{*}{A}^D=A^D{A}^{*}$, $A^{†}{A}^D=A^D{A}^{†}$, $A^{†}A{A}^D=A^{D}A{A}^{†}$, $A{A}^D{A}^{*}=A^{*}{A}^{D}A$ and so on to hold by using rank equalities of matrices. Some related topics are also examined.