Let $A$ be a square matrix. If $A$ satisfies the following two conditions

(1) $A$ is upper triangular

(2) all diagonal entries of $A$ are zero

then it is easy to see that all principal minors of $A$ vanish.

Now the problem I can not solve is

If all principal minors of a square matrix (not necessarily symmetric) vanish, show that it is permutation-similar to a matrix $A$ satisfying above two conditions.