Determine the characteristic polynomial of the matrix

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

For which values of $x \in \mathbb{C}$ is $M$ invertible? When $M$ is not invertible determine (i) the Jordan normal form $J$ of $M$, (ii) the minimal polynomial of $M$.

Find a basis of $\mathbb{C}^{4}$ such that $J$ is the matrix representing the endomorphism $M: \mathbb{C}^{4} \rightarrow \mathbb{C}^{4}$ in this basis. Give a change of basis matrix $P$ such that $P^{-1} M P=J$.