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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
$A^{2005}= \begin{pmatrix} 1 & 2005\\ 0& 1 \end{pmatrix}$
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Nice explaination :)
Is there any pattern to find the orthogonal property. It is not mentioned that matix is orthogonal. Do we need to check explicitly(by calculating P*P-transpose) its orthogonal  property or is there any tip to figure it out whether the matrix is orthogonal or not?