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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.