Find a non-diagonizable matrix that commutes with $\begin{bmatrix}0 & 0& -1\\1 & 1 &1\\0 & 0 & 1\end{bmatrix}$ I'm really lost on this one, with no clue on where to begin. Any help would be appreciated.

Not a general approach, but one solution is easy to see. You have for example
$$\begin{pmatrix} 0&0&0 \\ 0&0&1\\0&0&0\end{pmatrix}\cdot
\begin{pmatrix}0&0&-1\\1&1&1\\0&0&1\end{pmatrix} = \begin{pmatrix} 0&0&0 \\ 0&0&1\\0&0&0\end{pmatrix}=\begin{pmatrix}0&0&-1\\1&1&1\\0&0&1\end{pmatrix}\cdot\begin{pmatrix} 0&0&0 \\ 0&0&1\\0&0&0\end{pmatrix}$$
