Using the matrix determinant lemma, if $A$ is a $3 \times 3$ invertible matrix with $\det(A) = 4$, and $\det(A + \mathbf{u}\mathbf{v}^T) = \det(A)(1 + \mathbf{v}^T A^{-1} \mathbf{u})$, for what value of the scalar $\mathbf{v}^T A^{-1} \mathbf{u}$ does the rank-1 update produce a singular matrix?