Give a example of 2$\times$2 matrix A such that A has one independent eigenvector while $A^2$ has two independent eigenvectors.

Question

Give a example of 2$\times$2 matrix A such that A has one independent eigenvector while $A^2$ has two independent eigenvectors.

I do found a explaination here. here

And I followed it to make X as $\pmatrix{0&1\\1&0}$ and get the A as $\pmatrix{0&0\\1&0}$. Not really sure whether I am right.

Yeah, I checked that post and my problem is I still don't know how to get the A. Because it says x should be nonsingular so I tried $\pmatrix{0&1\\1&0}$ but it seems it's wrong..