For a counterexample to (2) with all $a_i$ nonzero, consider the $N=3$ case with $a_1=4, a_2=a_3=1$:
$$ \left[\matrix{0 & 1 & -1\cr -4 & 3 & 1\cr 4 & -1 & -3\cr}\right] \left[ \matrix{1 \cr 2\cr 0\cr}\right] = \left[\matrix{2\cr 2\cr 2\cr}\right]$$