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Let $A_1,A_2,\cdots ,A_n$ be complex matrices of the same size. We show in this note that the Moore-Penrose inverse, the Drazin inverse and the weighted Moore-Penrose inverse of the sum ${\sum }_{t=1}^n{A}_t$ can all be determined by the block circulant matrix generated by $A_1,A_2,\cdots ,A_n$. In addition, some equalities are also presented for the Moore-Penrose inverse and the Drazin inverse of a quaternionic matrix.

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.

It is shown that $$\text{rank}\left(P^{*}AQ\right)=\text{rank}\left(P^{*}A\right)+\text{rank}\left(AQ\right)-\text{rank}\left(A\right),$$ where $A$ is idempotent, $[P,Q]$ has full row rank and $P^{*}Q=0$. Some applications of the rank formula to generalized inverses of matrices are also presented.