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Dear student, A = [\begin{matrix} 1 & w \\ w^{2} & w \end{matrix}] A^{T} = [\begin{matrix} 1 & w^{2} \\ w & w \end{matrix}] {AA}^{T} = [\begin{matrix} 1 & w \\ w^{2} & w \end{matrix}] [\begin{matrix} 1 & w^{2} \\ w & w \end{matrix}] = [\begin{matrix} 1 + w^{2} & w^{2} + w^{2} \\ w^{2} + w^{2} & w^{4} + w^{2} \end{matrix}] w^{4} = w^{3} \times w = 1 \times w = w w^{2} + w + 1 = 0 So, 1 + w^{2} = - w; also, w^{2} + w = - 1 So, {AA}^{T} = [\begin{matrix} - w & 2 w^{2} \\ 2 w^{2} & w^{2} + w \end{matrix}] {AA}^{T} = [\begin{matrix} - w & 2 w^{2} \\ 2 w^{2} & - 1 \end{matrix}] Option D is correct . Regards

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