Let P=pmatrix√32 & 12 -12 & √32pmatrix, A=pmatrix1&10&1pmatrix, and Q=PAP^T.

Question

Let $P=\begin{pmatrix}\frac{\sqrt3}2 & \frac12\ -\frac12 & \frac{\sqrt3}2\end{pmatrix}$, $A=\begin{pmatrix}1&1\0&1\end{pmatrix}$, and $Q=PAP^T$. If $P^TQ^{2007}P=\begin{pmatrix}a&b\c&d\end{pmatrix}$, then $2a+b-3c-4d$ equals

Accepted Answer

$P^TP=I$ (orthogonal), so $P^TQ^{2007}P=P^T(PAP^T)^{2007}P=A^{2007}$ (telescoping).

$A^n=\begin{pmatrix}1&n\0&1\end{pmatrix}$, so $A^{2007}=\begin{pmatrix}1&2007\0&1\end{pmatrix}$: $a=1,\ b=2007,\ c=0,\ d=1$.

$2a+b-3c-4d=2+2007-0-4=2005$.

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