Let $\mathrm{\omega }$ be a complex cube root of unity with $\mathrm{\omega }\ne 1$ and P = [p_ij] be a n x n matrix with ${\mathrm{p}}_{\mathrm{ij}}={\mathrm{\omega }}^{\mathrm{i}+\mathrm{j}}$. Then ${\mathrm{P}}^2\ne \mathrm{O}$, when n =

$\mathrm{P}=\left[\begin{array}{ccccc}{\mathrm{\omega }}^2& {\mathrm{\omega }}^3& {\mathrm{\omega }}^4& \cdots & {\mathrm{\omega }}^{\mathrm{n}+1}\\ {\mathrm{\omega }}^3& {\mathrm{\omega }}^4& {\mathrm{\omega }}^5& \cdots & {\mathrm{\omega }}^{\mathrm{n}+2}\\ ⋮& ⋮& ⋮& ⋮& ⋮\\ {\mathrm{\omega }}^{\mathrm{n}+1}& {\mathrm{\omega }}^{\mathrm{n}+2}& \cdots & \cdots & {\mathrm{\omega }}^{2\mathrm{n}}\end{array}\right]$

or

${\mathrm{P}}^2=\left[\begin{array}{cccc}{\mathrm{\omega }}^4+{\mathrm{\omega }}^6\dots & {\mathrm{\omega }}^5+{\mathrm{\omega }}^7+\dots & \dots & \dots \\ {\mathrm{\omega }}^5+{\mathrm{\omega }}^7+\dots & \dots & \dots & \dots \\ ⋮& ⋮& ⋮& \dots \\ {\mathrm{\omega }}^{\mathrm{n}+2}+{\mathrm{\omega }}^{\mathrm{n}+4}\dots & \dots & \dots & {\mathrm{\omega }}^{2\mathrm{n}+2}+{\mathrm{\omega }}^{2\mathrm{n}+4}\dots \end{array}\right]$

     = Null matrix if n is a multiple of 3.