how to find $(I + uv^T)^{-1}$ Let $u, v \in \mathbb{R}^N, v^Tu \neq -1$. Then I know that $I +uv^T \in \mathbb{R}^{N \times N}$ is invertible and I can verify that $$(I + uv^T)^{-1} = I - \frac{uv^T}{1+v^Tu}.$$ But I am not able to derive that inverse on my own. How to find it actually? That is if am given $A =  I +uv^T$ and I am asked to find $A^{-1}$, how to get the answer?

If you guess that the inverse has a similar form,
$$(I+uv^T)^{-1}=\alpha I+\beta uv^T\ ,$$
then multiplying out and equating coefficients will solve the problem. 
