Let α be a root of the equation x2+x+1=0 and the matrix A=131111αα21α2α4 then the matrix A31 is equal to

x2+x+1=0 ⇒ α=ω or ω2A=131111ωω21ω2ω⇒A2=100001010⇒A4=I∴A31=A28.A3=IA3=A3

Let α be a root of the equation x2+x+1=0 and the matrix A=131111αα21α2α4 then the matrix A31 is equal to

x2+x+1=0 ⇒ α=ω or ω2A=131111ωω21ω2ω⇒A2=100001010⇒A4=I∴A31=A28.A3=IA3=A3

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Let $α$ be a root of the equation $x^{2} + x + 1 = 0$ and the matrix $A = \frac{1}{\sqrt{3}} [\begin{matrix} 1 & 1 & 1 \\ 1 & α & α^{2} \\ 1 & α^{2} & α^{4} \end{matrix}]$ then the matrix $A^{31}$ is equal to

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$A$

$A^{2}$

$A^{3}$

$I$

answer is C.

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Detailed Solution

$x^{2} + x + 1 = 0 \Rightarrow α = ω o r ω^{2}$

$A = \frac{1}{\sqrt{3}} [\begin{matrix} 1 & 1 & 1 \\ 1 & ω & ω^{2} \\ 1 & ω^{2} & ω \end{matrix}] \Rightarrow A^{2} = [\begin{matrix} 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{matrix}] \Rightarrow A^{4} = I$

$∴ A^{31} = A^{28} . A^{3} = (I) A^{3} = A^{3}$

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Let α be a root of the equation x2+x+1=0 and the matrix A=131111αα21α2α4 then the matrix A31 is equal to

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Let $α$ be a root of the equation $x^{2} + x + 1 = 0$ and the matrix $A = \frac{1}{\sqrt{3}} [\begin{matrix} 1 & 1 & 1 \\ 1 & α & α^{2} \\ 1 & α^{2} & α^{4} \end{matrix}]$ then the matrix $A^{31}$ is equal to