Given $P=\begin{pmatrix} \frac{\sqrt{3}}{2} & \frac{1}{2}\\ \frac{-1}{2} & \frac{\sqrt{3}}{2} \end{pmatrix}$
P is an orthogonal matrix because $P.P^{T}=I=P^{T}.P$
So, $Q^{2005} =(PAP^{T})(PAP^{T})(PAP^{T})....(PAP^{T})$ 2005 times.
Therefore, X be written as $P^{T} A^{2005} P = A^{2005}$
$A^{2}= \begin{pmatrix}
1 & 1\\
0& 1
\end{pmatrix} . \begin{pmatrix}
1 & 1\\
0& 1
\end{pmatrix} = \begin{pmatrix}
1 & 2\\
0& 1
\end{pmatrix}$
$A^{3}= \begin{pmatrix}
1 & 2\\
0& 1
\end{pmatrix} . \begin{pmatrix}
1 & 1\\
0& 1
\end{pmatrix} = \begin{pmatrix}
1 & 3\\
0& 1
\end{pmatrix}$
Similarly
$A^{2005}= \begin{pmatrix}
1 & 2005\\
0& 1
\end{pmatrix}$