LA2.1.4

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If w is cube root of unity,show that {{\begin{bmatrix}1&w&w^{2}\\w&w^{2}&1\\w^{2}&1&w\end{bmatrix}}+{\begin{bmatrix}w&w^{2}&1\\w^{2}&1&w\\w&w^{2}&1\end{bmatrix}}}{\begin{bmatrix}1\\w\\w^{2}\end{bmatrix}}={\begin{bmatrix}0\\0\\0\end{bmatrix}}\,

w is a cube root of unity implies 1+w+w^{2}=0,w^{3}=1\,

Now {{\begin{bmatrix}1&w&w^{2}\\w&w^{2}&1\\w^{2}&1&w\end{bmatrix}}+{\begin{bmatrix}w&w^{2}&1\\w^{2}&1&w\\w&w^{2}&1\end{bmatrix}}}{\begin{bmatrix}1\\w\\w^{2}\end{bmatrix}}={\begin{bmatrix}1+w&w+w^{2}&w^{2}+1\\w+w^{2}&w^{2}+1&1+w\\w^{2}+w&1+w^{2}&w+1\end{bmatrix}}{\begin{bmatrix}1\\w\\w^{2}\end{bmatrix}}\,

{\begin{bmatrix}-w^{2}&-1&-w\\-1&-w&-w^{2}\\-1&-w&-w^{2}\end{bmatrix}}{\begin{bmatrix}1\\w\\w^{2}\end{bmatrix}}\,

{\begin{bmatrix}-w^{2}-w-w^{3}\\-1-w^{2}-w^{4}\\-1-w^{2}-w^{4}\end{bmatrix}}={\begin{bmatrix}-w^{2}-w-1\\-1-w^{2}-w\\-1-w^{2}-w\end{bmatrix}}={\begin{bmatrix}0\\0\\0\end{bmatrix}}\,


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