(Linear Algebra) Distinct Eigenvalues of a Matrix

[Gg199]

I am reading through a proof and one line of it is not immediately obvious to me, despite it's simplicity. It relates to eigenvalues of a (nearly) full rank, symmetric matrix.

Say we have a symmetric matrix A(nxn) that has rank=n-1. Why is this enough to say that all eigenvalues of A are distinct? Note that the symmetry is important for the result to hold, but I don't understand why.

Thank you in advance.

 

[AlephZero]

Something doesn't make sense here. What about
$$\begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{pmatrix}$$

If the rank is n-1, the only thing you can say is that exactly one eigenvalue is zero.

 

[Gg199]

Sorry I didn't give enough details at all. I think I understand it now though, thank you for the help anyway.