Integrals I: Feynman's Trick · Superchallenge problem

Let $\mathbf u = (1, 2, 3)^{\mathsf T}$ and $\mathbf v = (4, 5, 6)^{\mathsf T}$. Compute $\det\!\left(I + \mathbf u\mathbf v^{\mathsf T}\right)$.