Finding if symmetric matrix exists given eigenvectors

Question

I'm trying to solve this problem below:

I'm kinda lost on how to approach the problem. The only thing I can note is that the latter two eigenvectors are linear combinations of the first one and

\begin{bmatrix} 1 \\ 1 \\ 1 \\ \end{bmatrix}

Not really sure if that's relevant at all or just a coincidence. Any help would be great!

Answer 1

if $A$ is symmetric, $Av= \lambda v$ and $Aw = \mu w$ and $0 \neq\lambda- \mu $ then $$\lambda \langle v, w\rangle = \langle Av, w\rangle = \langle v, Aw\rangle= \mu \langle v, w\rangle$$

so ...? Can you finish with this hint?