The meaning of $ A {\bf 1} = 0 $ and $ A^T {\bf 1} = 0 $

Question

I have a square matrix A that its columns and rows entries sum to zero namely :

$ A {\bf 1} = 0 $
$ A^T {\bf 1} = 0 $

$ {\bf 1} $ is the all ones vector.

I tried to prove that A is symmetric but got stuck and on the other could not come with a counter example. What can I say about A with such properties , it definitely has a common eigenvector with its transpose but could not get much more.

Thanks!

Answer 1

A simple counterexample: $$A=\begin{bmatrix} 1 & -1 & 0 \\ 0 & 0 & 0 \\ -1 & 1 & 0 \\ \end{bmatrix}$$