For given a real matrix $A$ of size $n$ by $n$, we assume that there exist $\lambda$ with multiplicity $2$ and its corresopnding eigenvector is denoted by $x$. In this situation, prove that
if the dimension of the null space of $A-\lambda I$ is $1$, then $x$ belongs to the column space of $A-\lambda I$.
Actually, I tried many times, but I failed to prove this. Can someone let me know the proof? Some hints is also very thank you?