Today I discovered this question: Determinant of rank-one perturbations of (invertible) matrices
There, the OP asks how the determinant of $I + vv^T$ can be computed. Nevertheless, (one of) the answers handle(s) the more general question of $\det (A + cd^T)$.
Is it OK to edit the question to ask for this directly?
My main motivation for this is that this would make searching (and finding) the nice answer easier.
Concretely, I was looking for the $\det (A + cd^T)$ case and almost closed the question after reading that the OP only asked for $\det (I + vv^T)$. Luckily, I read on.