If $P=\begin{bmatrix}\frac{\sqrt{3}}{2}&\frac{1}{2}\\-\frac{1}{2}&\frac{\sqrt{3}}{2}\end{bmatrix},A=\begin{bmatrix}1&1\\0&1\end{bmatrix}$ and $Q = PAP^T$, then $p^T (Q^{2005})p$ is equal to

$\begin{bmatrix}1&2005\\0&1\end{bmatrix}$

$∵ Q = PAP^T ⇒ P^T Q = AP^T (∵PP^T = I)$

$∴P^T\, Q^{2005} P = AP^T\,Q^{2004} P$

$= A^2\, P^T\, Q^{2003} P = A^3\, P^T\, Q^{2002} P = ....$

$= A^{2004} P^T (QP) = A^{2004} P^T (PA)$

$(∵ Q = PAP^T ⇒ QP = PA)$

$=A^{2005}=\begin{bmatrix}1&2005\\0&1\end{bmatrix}$