# Prove the following

Question:

If $P=\left[\begin{array}{ll}1 & 0 \\ 1 / 2 & 1\end{array}\right]$, then $P^{50}$ is:

1. $\left[\begin{array}{ll}1 & 0 \\ 25 & 1\end{array}\right]$

2. $\left[\begin{array}{ll}1 & 50 \\ 0 & 1\end{array}\right]$

3. $\left[\begin{array}{ll}1 & 25 \\ 0 & 1\end{array}\right]$

4. $\left[\begin{array}{ll}1 & 0 \\ 50 & 1\end{array}\right]$

Correct Option: 1

Solution:

$P=\left[\begin{array}{ll}1 & 0 \\ \frac{1}{2} & 1\end{array}\right]$

$\mathrm{P}^{2}=\left[\begin{array}{ll}1 & 0 \\ \frac{1}{2} & 1\end{array}\right]\left[\begin{array}{ll}1 & 0 \\ \frac{1}{2} & 1\end{array}\right]=\left[\begin{array}{ll}1 & 0 \\ 1 & 1\end{array}\right]$

$\mathrm{P}^{3}=\left[\begin{array}{ll}1 & 0 \\ 1 & 1\end{array}\right]\left[\begin{array}{ll}1 & 0 \\ \frac{1}{2} & 1\end{array}\right]=\left[\begin{array}{ll}1 & 0 \\ \frac{3}{2} & 1\end{array}\right]$

$\mathrm{P}^{4}=\left|\begin{array}{ll}1 & 0 \\ 1 & 1\end{array}\right|\left|\begin{array}{ll}1 & 0 \\ 1 & 1\end{array}\right|=\left|\begin{array}{ll}1 & 0 \\ 2 & 1\end{array}\right|$

$\therefore \mathrm{P}^{50}=\left[\begin{array}{cc}1 & 0 \\ 25 & 1\end{array}\right]$