Here, $A = [a_{ij}]_{n \times n}$ and $a_{ij} = \omega^{i+j}$
When $n = 1$
$A = [a_{ij}]_{1 \times 1} = [\omega^{2}]$
$\implies A^{2} = [\omega^{2}]^{2} = [\omega^{4}] = [\omega] \neq O \quad [\because \omega ^{3} = 1]$
When $n = 2$
$A = [a_{ij}]_{2 \times 2} = \begin{bmatrix} \omega^{2} & \omega^{3} \\ \omega^{3} & \omega^{4} \end{bmatrix} = \begin{bmatrix} \omega^{2} & 1 \\ 1 & \omega \end{bmatrix}$
$ \implies A^{2} = \begin{bmatrix} \omega^{2} & 1 \\ 1 & \omega \end{bmatrix}\begin{bmatrix} \omega^{2} & 1 \\ 1 & \omega \end{bmatrix} = \begin{bmatrix} \omega^{4} + 1 & \omega^{2} + \omega \\ \omega^{2} + \omega & 1 + \omega^{2} \end{bmatrix} \neq O$
When $n = 3$
$A = [a_{ij}]_{3 \times 3} = \begin{bmatrix} \omega^{2} & \omega^{3} & \omega^{4} \\ \omega^{3} & \omega^{4} & \omega^{5} \\ \omega^{4} & \omega^{5} & \omega^{6} \end{bmatrix} = \begin{bmatrix} \omega^{2} & 1 & \omega \\ 1 & \omega & \omega^{2} \\ \omega & \omega^{2} & 1 \end{bmatrix}$
$\implies A^{2} = \begin{bmatrix} \omega^{2} & 1 & \omega \\ 1 & \omega & \omega^{2} \\ \omega & \omega^{2} & 1 \end{bmatrix} \begin{bmatrix} \omega^{2} & 1 & \omega \\ 1 & \omega & \omega^{2} \\ \omega & \omega^{2} & 1 \end{bmatrix}$
$\implies A^{2} = \begin{bmatrix} \omega^{4} + 1 + \omega^{2} & \omega^{2} + \omega + \omega^{3} & \omega^{3} + \omega^{2} + \omega \\ \omega^{2} + \omega + \omega^{3} & 1 + \omega^{2} + \omega^{4} & \omega + \omega^{3} + \omega^{2} \\ \omega^{3} + \omega^{2} + \omega & \omega + \omega^{3} + \omega^{2} & \omega^{2} + \omega^{4} + 1 \end{bmatrix}$
$\implies A^{2} = \begin{bmatrix} \omega + 1 + \omega^{2} & \omega^{2} + \omega + 1 & 1 + \omega^{2} + \omega \\ \omega^{2} + \omega + 1 & 1 + \omega^{2} + \omega & \omega + 1 + \omega^{2} \\ 1 + \omega^{2} + \omega & \omega + 1 + \omega^{2} & \omega^{2} + \omega + 1 \end{bmatrix}\qquad [\because \omega ^{3} = 1]$
$\implies A^{2} = \begin{bmatrix} 0 & 0 & 0 \\
0 & 0 & 0 \\
0 & 0 & 0 \\
\end{bmatrix}_{3 \times 3}\qquad [\because 1 + \omega + \omega ^{2} = 0]$
$\therefore$ When $n$ is multiple of $3,A^{2} = O,$ otherwise $A^{2} \neq O.$
So, the correct answer is $A;C;D.$
