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Properties of Determinants

134. Prove th...

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134. Prove that (1+w)-(1-w) =0 , where w is cube root of unity

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1, w, w^{2}

are the cube roots of unity, prove that

{(1 - w + w^{2})}^{6} + {(1 - w^{2} + w)}^{6} = 128

\frac{(a + b w + c w^{2})}{(c + a w + b w^{2})} = w^{2}

w

being cube root of unity.

∣ ∣ ∣ ∣ ∣ \begin{matrix} 1 & w & w^{2} w & w^{2} & 1 w^{2} & w & 1 \end{matrix} ∣ ∣ ∣ ∣ ∣

Where w is a complex cube root of unity.

w (\neq 1)

is a cube root of unity, then

∣ ∣ ∣ ∣ ∣ \begin{matrix} 1 & 1 + i + w^{2} & w^{2} 1 - i & - 1 & w^{2} - 1 - i & - i + w - 1 & - 1 \end{matrix} ∣ ∣ ∣ ∣ ∣

If 1, w, $w^{2}$ are three cube roots of unity, then $(1 - w + w^{2}) (1 + w - w^{2})$ is .............

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