Let $\omega$ be a complex cube root of unity with $\omega \neq 1$ and $P = [p_{ij}]$ be a $n \times n$ matrix with $p_{ij} = \omega^{i+j}$. Then $p^{2} \neq 0$, when $n =$

$\omega^{3}=1, \quad \omega\neq 1, \quad 1+\omega+\omega^{2}=0$ $P=[p_{ij}], \quad p_{ij}=\omega^{i+j}, \quad 1\le i,j\le n$ $\text{Find } n \text{ such that } P^{2}\neq 0$ $(1)\ 57, \quad (2)\ 55, \quad (3)\ 58, \quad (4)\ 56$ $1.\ \text{structure of } P:\ p_{ij}=\omega^{i+j}$ $2.\ (P^{2})_{ij} \text{ from the matrix product}$ $3.\ P^{2}=\lambda P \Rightarrow \text{condition for } P^{2}=0$ $p_{ij}=\omega^{i+j}=\omega^{i}\,\omega^{j}$ $P=uv^{T}, \quad u_i=\omega^{i}, \ v_j=\omega^{j}$ $(P^{2})_{ij}=\sum_{k=1}^{n} p_{ik}\,p_{kj}$ $=\sum_{k=1}^{n}\omega^{i+k}\,\omega^{k+j}$ $=\sum_{k=1}^{n}\omega^{i+j+2k}=\omega^{i+j}\sum_{k=1}^{n}\omega^{2k}$ $(P^{2})_{ij}=\lambda\,p_{ij} \Rightarrow P^{2}=\lambda P$ $\lambda=\sum_{k=1}^{n}\omega^{2k}; \quad P\neq 0 \Rightarrow (P^{2}=0 \iff \lambda=0)$ $\{\omega^{2k}\}: \ \omega^{2},\ \omega^{4}=\omega,\ \omega^{6}=1,\ \omega^{2},\dots$ $\omega^{2}+\omega+1=0 \Rightarrow \text{every three terms sum to } 0$ $n\equiv 0\ (\mathrm{mod}\ 3):\ \lambda=0$ $n\equiv 1:\ \lambda=\omega^{2}; \qquad n\equiv 2:\ \lambda=\omega^{2}+\omega=-1$ $P^{2}=0 \iff 3 \mid n$ $57=3\times 19; \quad 55,58,56 \not\equiv 0\ (\mathrm{mod}\ 3)$ $\boxed{n=57 \quad (\text{option }1)}$