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If $P = [\begin{matrix} \frac{\sqrt{3}}{2} & \frac{1}{2} \\ \frac{- 1}{2} & \frac{\sqrt{3}}{2} \end{matrix}]$ , $A = [\begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix}]$ , $Q = P A P^{T}$ and $P^{T} Q^{2012} P = [\begin{matrix} α & γ \\ β & α \end{matrix}]$ then the value of $α + β$ is

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answer is C.

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Detailed Solution

$Q^{2} = P A P^{T} \times P A P^{T}$
$= P A (P^{T} P) A P^{T} ∵ P P^{T} = P^{T} P = I$
$= P A^{2} P^{T}$
$Q^{2012} = P A^{2012} P^{T}$
$P^{T} Q^{2012} P = A^{2012} = [\begin{matrix} 1 & 2012 \\ 0 & 1 \end{matrix}]$
$α = 1, β = 0 \to α + β = 1$

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