Existence of matrices with non-zero principal minors The problem sounds very simple but I have yet to come to an answer.
Prove or disprove:

For all $n$ there exists a matrix $A \in \mathbb{R}^{n \times n}$ with $\det(A) = 0$ such that all first minors $M_{i,j} \neq 0$.

Trivial example for $n=2$:
\begin{pmatrix}
1 & 1\\
1 & 1
\end{pmatrix}

Hint: assume the $n$ variable matrix looks the same (all ones), and compute it's determinant by the cofactor of minors method. We know what the determinant of any possible 2 by 2 minor is. What does this tell us? 
