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If $P=\left[\begin{array}{ll}1 & 0 \\ 1 / 2 & 1\end{array}\right]$, then $P^{50}$ is:
Correct Option: 1
$P=\left[\begin{array}{ll}1 & 0 \\ \frac{1}{2} & 1\end{array}\right]$
$\mathrm{P}^{2}=\left[\begin{array}{ll}1 & 0 \\ \frac{1}{2} & 1\end{array}\right]\left[\begin{array}{ll}1 & 0 \\ \frac{1}{2} & 1\end{array}\right]=\left[\begin{array}{ll}1 & 0 \\ 1 & 1\end{array}\right]$
$\mathrm{P}^{3}=\left[\begin{array}{ll}1 & 0 \\ 1 & 1\end{array}\right]\left[\begin{array}{ll}1 & 0 \\ \frac{1}{2} & 1\end{array}\right]=\left[\begin{array}{ll}1 & 0 \\ \frac{3}{2} & 1\end{array}\right]$
$\mathrm{P}^{4}=\left|\begin{array}{ll}1 & 0 \\ 1 & 1\end{array}\right|\left|\begin{array}{ll}1 & 0 \\ 1 & 1\end{array}\right|=\left|\begin{array}{ll}1 & 0 \\ 2 & 1\end{array}\right|$
$\therefore \mathrm{P}^{50}=\left[\begin{array}{cc}1 & 0 \\ 25 & 1\end{array}\right]$
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