Elements of a matrix A of order $\text{10\hspace{0.17em}\hspace{0.17em}}\times \text{\hspace{0.17em}10}$ are defined as $a_{ij}={\omega }^{i+j}$ (where $\omega$ is imaginary cube root of unity), then trace (A) of the matrix is

$$\begin{gathered} tr\left(A\right)=\sum _{i=j}{a}_{ij} \\ =(a_{11}+a_{22}+a_{33}+\mathrm{...}+a_{1010}) \\ =({\omega }^2+{\omega }^4+{\omega }^6+\mathrm{...}+{\omega }^{20}) \\ ={\omega }^2(1+{\omega }^2+{\omega }^4+\mathrm{...}+{\omega }^{18}) \\ ={\omega }^2[(1+\omega +{\omega }^2)+\mathrm{...}+(1+\omega +{\omega }^2)+1] \\ ={\omega }^2\times 1 \\ ={\omega }^2 \end{gathered}$$