Let $A \in \mathrm{GL}_n(\mathbb{R})$ be an invertible matrix, and let $\mathbf{u}, \mathbf{v} \in \mathbb{R}^n$. Suppose that $A + \mathbf{u v}^T$ is invertible. Show that
$$\left(A + \mathbf{u v}^T\right)^{-1} = A^{-1} - \frac{A^{-1} \mathbf{u v}^T A^{-1}}{1 + \left\langle \mathbf{v}, A^{-1} \mathbf{u} \right\rangle}.$$