Let $A \in \mathrm{GL}_n(\mathbb{R})$ be an invertible matrix, and let $\mathbf{u}, \mathbf{v} \in \mathbb{R}^n$. Calculate the block matrix product
$$\left(\begin{array}{cc} \mathbb{I}_n & 0 \\ \mathbf{v}^T & 1 \end{array}\right) \left(\begin{array}{cc} \mathbb{I}_n + \mathbf{u v}^T & \mathbf{u} \\ 0 & 1 \end{array}\right) \left(\begin{array}{cc} \mathbb{I}_n & 0 \\ -\mathbf{v}^T & 1 \end{array}\right).$$