Eigenvalues · Challenge problem

Let $\mathbf{u} = \begin{pmatrix}1\\2\\3\end{pmatrix}$ and $\mathbf{v} = \begin{pmatrix}4\\5\\6\end{pmatrix}$. Find $\det\left(I + \mathbf{u}\mathbf{v}^T\right)$, where $I$ is the $3\times3$ identity, by finding the eigenvalues of $\mathbf{u}\mathbf{v}^T$.