Eigenvector standardization

I have matrix

and I get that

so

**//**after this our lecturer takes random numbers and checks if they can be eigenvalues.

For this problem I'm sure that eigenvalue is only

**//**but what if I get ?

**//**I have to check all numbers till 6

**//**so how did he find that for this one is and so fast?

**//**** EH... NEVERMIND xD**

further, to find eigenvector I have

but I can't find a way to substitute so that I would have 2 even rows or columns, so maybe there is another way I don't know?

**some update:**

I ended up at

So, after getting someone annoying for updating my post so many times xD

**Q: HOW DO I STANDARDIZE EIGENVECTOR AND CHECK IF ANSWER IS CORRECT?**

someone told me: "you need to find vector length or smth" =/

???????

IF CORRECT, HOW DO I CHECK IF IT IS SO?