Let $\omega$ be a complex cube root of unity with $\omega \ne 1\text{ and }\mathrm{P}=\left[P_{ij}\right]$ be a n × n matrix with  ${\mathrm{P}}_{\mathrm{ij}}={\mathrm{\omega }}^{\mathrm{i}+\mathrm{j}}$ . Then $P^2\ne 0$ , when n =

Here $P={\left[p_{ij}\right]}_{m\times n}\text{ with }{p}_{ij}={\omega }^{i+j}$
$P={\left[p_{ij}\right]}_{1\times 1}=\left[{\omega }^2\right]\Rightarrow P^2=\left[{\omega }^4\right]\ne 0$
$\therefore P^2=0$, when n is multiple of 3.
$P^2\ne 0$, when n is not a multiple of 3.
n=57 is not possible  
$\therefore$n=55,58,56 are possible.