Determine for which values of $x \in \mathbb{C}$ the matrix

$$M=\left(\begin{array}{ccc} x & 1 & 1 \\ 1-x & 0 & -1 \\ 2 & 2 x & 1 \end{array}\right)$$

is invertible. Determine the rank of $M$ as a function of $x$. Find the adjugate and hence the inverse of $M$ for general $x$.