Let $I$ be a $n$ x $n$ matrix, and $v$ and $w$ a nonzero $n$ x $1$ column matrix. Imagine $vw^T$ is nonsingular (therefore $(vw^T)^{-1}$ exist). Here is my prove:
$(I+vw^T)^{-1}=I-\dfrac{vw^T}{1+w^Tv} \Rightarrow$
$I=(I+vw^T)(I+vw^T)^{-1}=(I+vw^T)(I-\dfrac{vw^T}{1+w^Tv})=II+I(-\dfrac{vw^T}{1+w^Tv})+vw^TI+vw^T(-\dfrac{vw^T}{1+w^Tv})=I-\dfrac{vw^T}{1+w^Tv}+vw^T-\dfrac{vw^Tvw^T}{1+w^Tv} \Rightarrow$
$0=vw^T-\dfrac{vw^Tvw^T}{1+w^Tv}-\dfrac{vw^T}{1+w^Tv} \Rightarrow$ $vw^T=\dfrac{vw^T}{1+w^Tv}(1+vw^T) \Rightarrow$ $(1+w^Tv)vw^T=vw^T(1+vw^T) \Rightarrow$ $vw^T(1+w^Tv)=vw^T(1+vw^T)$ because $vw^T$ is nonsingular: $1+w^Tv=1+vw^T$ therefore $w^Tv=vw^T$
Were am I wrong?