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 ${A}^{*}{A}^{D}={A}^{D}{A}^{*}$, ${A}^{\u2020}{A}^{D}={A}^{D}{A}^{\u2020}$, ${A}^{\u2020}A{A}^{D}={A}^{D}A{A}^{\u2020}$, $A{A}^{D}{A}^{*}={A}^{*}{A}^{D}A$ 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.

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