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