If P=pmatrix√32&12-12&√32pmatrix, A=pmatrix1&10&1pmatrix, and Q=PAP^T, then P^TQ^2015P is:

Question

If $P=\begin{pmatrix}\frac{\sqrt3}2&\frac12\-\frac12&\frac{\sqrt3}2\end{pmatrix}$, $A=\begin{pmatrix}1&1\0&1\end{pmatrix}$, and $Q=PAP^T$, then $P^TQ^{2015}P$ is:

Accepted Answer

$PP^T=P^TP=I$, so $Q^n=PA^nP^T$ (telescoping), and $P^TQ^{2015}P=A^{2015}$.

$A^n=\begin{pmatrix}1&n\0&1\end{pmatrix}$, so $A^{2015}=\begin{pmatrix}1&2015\0&1\end{pmatrix}$.

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