If  $P=\left[\begin{array}{cc}\frac{\sqrt{3}}{2}& \frac{1}{2}\\ \frac{-1}{2}& \frac{\sqrt{3}}{2}\end{array}\right] ,A=\left[\begin{array}{cc}1& 1\\ 0& 1\end{array}\right],Q=PA{P}^T$ and if  $P^T{Q}^{2012}P=\left[\begin{array}{cc}\alpha & \gamma \\ \beta & \alpha \end{array}\right]$, then the value of  $(\alpha +\beta )$ is

$$\begin{gathered} Q^{2012}=\left(PA{P}^T\right)\left(PA{P}^T\right)\mathrm{......}\left(PA{P}^T\right)\left(2012times\right) \\ =PAA{P}^T\mathrm{.........}=P{A}^{2012}{P}^T \\ {P}^{T}P{A}^{2012}{P}^{T}P=A^{2012} but \\ {A}^2=\left[\begin{array}{cc}1& 1\\ 0& 1\end{array}\right]\left[\begin{array}{cc}1& 1\\ 0& 1\end{array}\right]=\left[\begin{array}{cc}1& 2\\ 0& 1\end{array}\right] \\ {A}^3=\left[\begin{array}{cc}1& 2\\ 0& 1\end{array}\right]\left[\begin{array}{cc}1& 1\\ 0& 1\end{array}\right]=\left[\begin{array}{cc}1& 3\\ 0& 1\end{array}\right] \\ \therefore A^{2012}=\left[\begin{array}{cc}1& 2012\\ 0& 1\end{array}\right];\therefore \alpha =1,\beta =0 \end{gathered}$$