Consider a matrix $U$ such that
$U = \left[\begin{array}{rrrrr} 1 & 1 & 1 &1&1\\ 1 &o & o^2 &o^3 & o^4\\ 1 & o^2 & o^4& o &o^3 \\ 1 & o^3 & o& o^4 & o^2\\ 1 & o^4 & o^3 &o^2 &o \\ \end{array}\right]$, where $1+o+o^2+o^3+o^4=0$.
Prove that if $(i,j)$ entry $u_{ij}$ of $U$ is same as $(k,l)$ entry $u_{kl}$ of $U$, for any power $U^n$ of $U$ $(i,j)$th entry and $(k,l)$th entry will be same.