If P = begin{bmatrix} frac{sqrt{3}}{2} & frac | vedclass

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(C) Given $P = \begin{bmatrix} \frac{\sqrt{3}}{2} & \frac{1}{2} \ -\frac{1}{2} & \frac{\sqrt{3}}{2} \end{bmatrix}$.Since $P$ is an orthogonal matrix,

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$\begin{bmatrix} 1 & 2015 \ 0 & 1 \end{bmatrix}$

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(C) Given $P = \begin{bmatrix} \frac{\sqrt{3}}{2} & \frac{1}{2} \ -\frac{1}{2} & \frac{\sqrt{3}}{2} \end{bmatrix}$.Since $P$ is an orthogonal matrix,$P^T P = P P^T = I$.Given $Q = PAP^T$,we need to find $P^T Q^{2015} P$.$Q^2 = (PAP^T)(PAP^T) = PA(P^T P)AP^T = PA(I)AP^T = PA^2 P^T$.By induction,$Q^n = PA^n P^T$.Therefore,$Q^{2015} = PA^{2015} P^T$.Now,$P^T Q^{2015} P = P^T (PA^{2015} P^T) P = (P^T P) A^{2015} (P^T P) = I A^{2015} I = A^{2015}$.Given $A = \begin{bmatrix} 1 & 1 \ 0 & 1 \end{bmatrix}$,we observe the pattern:$A^2 = \begin{bmatrix} 1 & 1 \ 0 & 1 \end{bmatrix} \begin{bmatrix} 1 & 1 \ 0 & 1 \end{bmatrix} = \begin{bmatrix} 1 & 2 \ 0 & 1 \end{bmatrix}$.$A^3 = \begin{bmatrix} 1 & 2 \ 0 & 1 \end{bmatrix} \begin{bmatrix} 1 & 1 \ 0 & 1 \end{bmatrix} = \begin{bmatrix} 1 & 3 \ 0 & 1 \end{bmatrix}$.By induction,$A^n = \begin{bmatrix} 1 & n \ 0 & 1 \end{bmatrix}$.Thus,$A^{2015} = \begin{bmatrix} 1 & 2015 \ 0 & 1 \end{bmatrix}$.

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 find $P^T Q^{2015} P$.

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