Fix Me or Crush Me

Imagine that you have a pair of vectors ${\bf F}$ and ${\bf Z}$
$$
{\bf F}=\pmatrix{1\cr 1 \cr 0}\quad {\bf Z}=\pmatrix{0\cr 1 \cr 1}
$$
Can you construct an example of a matrix $M$, other than the identity, which leaves ${\bf F}$ fixed, in that $M{\bf F}={\bf F}$? How many such matrices can you find? Which is the simplest? Which is the most complicated?

Can you construct an example of a matrix $N$, other than the zero matrix, which crushes ${\bf Z}$ to the zero vector ${\bf 0}$, in that $N{\bf Z}={\bf 0}$? How many such matrices can you find? Which is the simplest? Which is the most complicated?

Can you find a matrix which leaves ${\bf F}$ fixed and also crushes ${\bf Z}$?

Can you find any (many?) vectors fixed or crushed by the following matrices? Give examples or convincing arguments if no such vectors exist.
$$
M = \begin{pmatrix} 1&0&0\\ 0&1&0\\ 0&0&1\\ \end{pmatrix}, \begin{pmatrix} 1&2&3\\ 2&3&4\\ 3&4&5\\ \end{pmatrix}, \begin{pmatrix} 1&-2&1\\ 1&1&0\\ -2&1&-2\\ \end{pmatrix}
$$


Very hard extension: Imagine that you are given a vector ${\bf F}$ and a vector ${\bf Z}$. Investigate whether you will be able to make a matrix $M$ which both fixes ${\bf F}$ and crushes ${\bf Z}$.