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

 ${\mathrm{p}}^2=\left[\begin{array}{l}\begin{array}{cccc}{\mathrm{\omega }}^2& {\mathrm{\omega }}^3& ...& {\mathrm{\omega }}^{\mathrm{n}+1}\end{array}\\ \begin{array}{cccc}{\mathrm{\omega }}^3& {\mathrm{\omega }}^4& ...& ...\end{array}\\ \begin{array}{cccc}⋮& & ...& ...\end{array}\\ \begin{array}{cccc}{\mathrm{\omega }}^{\mathrm{n}+1}& {\mathrm{\omega }}^{\mathrm{n}+2}& ...& {\mathrm{\omega }}^{2\mathrm{n}}\end{array}\end{array}\right]\left[\begin{array}{l}\begin{array}{cccc}{\mathrm{\omega }}^2& {\mathrm{\omega }}^3& ...& {\mathrm{\omega }}^{\mathrm{n}+1}\end{array}\\ \begin{array}{cccc}{\mathrm{\omega }}^3& ...& ...& ...\end{array}\\ \begin{array}{cccc}⋮& & ...& ...\end{array}\\ \begin{array}{cccc}{\mathrm{\omega }}^{\mathrm{n}+1}& ...& ...& {\mathrm{\omega }}^{2\mathrm{n}}\end{array}\end{array}\right]=\left[\begin{array}{l}{\mathrm{\omega }}^4+{\mathrm{\omega }}^6+{\mathrm{\omega }}^8+........{\mathrm{\omega }}^{2\mathrm{n}+2}..............\\ ⋮\\ ⋮\\ ⋮\end{array}\right]$

for null matrix

            ω⁴ + ω⁶ + ω⁸ +.......... + ω^{2n + 2} must be zero

            and   in this case each elements and matrix p² will be zero

            so        ω⁴ + ω⁶ + ω⁸ +.......... + ω^{2n + 2} have n terms                                  .......(i)

            and      ω⁴ + ω⁶ + ω⁸ = ω + 1 + ω² = 0

                        ω¹⁰ + ω¹² + ω¹⁴ = 0

            i.e.        ω^{2n + 2} must gives ω²ⁿ · ω² 

            (Last term each triplets in ω²)

            $\therefore$         2n = 3k (k is an integer)

            $\therefore$         n must be divisible by 3 for P² to be null matrix

            $\therefore$         n = 57