Elements of a matrix $$A$$ of order $$10 \times 10$$  are defined as $$a_{ij} = w^{i+j}$$  (where $$w$$ is cube root of unity), then $$tr\left( A \right)$$  of the matrix is

Solution :
$$\eqalign{ & tr\left( A \right) = \sum\limits_{i = j} {{a_{ij}}} = \left( {{a_{11}} + {a_{22}} + {a_{33}} + ..... + {a_{10}}} \right) \cr & = \left( {{w^2} + {w^4} + {w^6} + ..... + {w^{20}}} \right) = {w^2}\left( {1 + {w^2} + {w^4} + ..... + {w^{18}}} \right) \cr & = {w^2}\left[ {\left( {1 + w + {w^2}} \right) + ..... + \left( {1 + w + {w^2}} \right) + 1} \right] = {w^2} \times 1 = {w^2} \cr} $$