In my answer to this question, there appears the following sub-question about a sum of roots of unity. Denoting $z=\exp\frac{i\pi}N$ (so that $z^N=-1$), can the quantity $$\sum_{k=0}^{N-1}z^{2k^2+k}$$ vanish ?

The answer is clearly No for $N\le7$. I suspect that it never vanishes, which should answer definitively the above matricial question. Any idea of a general argument ?