Given Matrix transformation matrix A=(111022003) Are there vectors in R3 that do not change under...

Step 1
Let
$u=\left(\begin{array}{c}\lambda \\ 0\\ 0\end{array}\right)$
for any ${\displaystyle \lambda }$
A little work shows that these are the only solutions to
${\displaystyle Au=u}$
A note: no matter what linear transformation you have, ${\displaystyle T\left(0\right)=0}$ is always true, but sometimes students forget to state the whole problem, so I posted a complete set of solutions just in case

Step 1
$\left(\begin{array}{ccc}1& 1& 1\\ 0& 2& 2\\ 0& 0& 3\end{array}\right)\left(\begin{array}{c}x\\ y\\ z\end{array}\right)=\left(\begin{array}{c}x\\ y\\ z\end{array}\right)⟺\left(\begin{array}{c}x+y+z\\ 2y+2z\\ 3z\end{array}\right)$
$=\left(\begin{array}{c}x\\ y\\ z\end{array}\right)⟺\left(\begin{array}{c}y+z\\ y+2z\\ 2z\end{array}\right)=\left(\begin{array}{c}0\\ 0\\ 0\end{array}\right).$
Third equation gives ${\displaystyle z=0}$ and first equation gives ${\displaystyle y=0}$. So,
$\left(\begin{array}{ccc}1& 1& 1\\ 0& 2& 2\\ 0& 0& 3\end{array}\right)\left(\begin{array}{c}x\\ 0\\ 0\end{array}\right)=\left(\begin{array}{c}x\\ 0\\ 0\end{array}\right)$
for any ${\displaystyle x\in \mathbb{R}.}$