Let $M = \begin{pmatrix} 3 & -1 & 0 \\ 4 & -1 & 0 \\ -2 & 1 & 1 \end{pmatrix}$. Without using direct expansion, we can write $M = I_3 + u v^T$ for some vectors $u, v \in \mathbb{R}^3$. What is the determinant of $M$?