For a counterexample to (1) with all $a_i$ nonzero, consider the $N=4$ case with $a_3 = a_1$, noting that $$ \left[\matrix{a_4-a_2 & a_2 & 0 & -a_4\cr -a_1 & 0 & a_1 & 0\cr 0 & -a_2 & a_2-a_4 & a_4\cr a_1 & 0 & -a_1 & 0\cr}\right] \left[\matrix{0 \cr a_4 \cr 0 \cr a_2}\right] = 0$$
For a counterexample to (2) with all $a_i$ nonzero, consider the $N=3$ case with $a_1=4, a_2=a_3=1$: $$ \left[\matrix{0 & 1 & -1\cr -4 & 3 & 1\cr 4 & -1 & -3\cr}\right] \left[ \matrix{1 \cr 2\cr 0\cr}\right] = \left[\matrix{2\cr 2\cr 2\cr}\right]$$