Question

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

• A. $\left[\begin{array}{cc}1& 25\\ 0& 1\end{array}\right]$
• B. $\left[\begin{array}{cc}1& 0\\ 50& 1\end{array}\right]$
• C. $\left[\begin{array}{cc}1& 0\\ 25& 1\end{array}\right]$
• D. $\left[\begin{array}{cc}1& 50\\ 0& 1\end{array}\right]$

Answer 1

The correct option is C $\left[\begin{array}{cc}1& 0\\ 25& 1\end{array}\right]$

Given: $P=\left[\begin{array}{cc}1& 0\\ 1/2& 1\end{array}\right]$
$\Rightarrow P^2=\left[\begin{array}{cc}1& 0\\ 1/2& 1\end{array}\right]\left[\begin{array}{cc}1& 0\\ 1/2& 1\end{array}\right]$
$\therefore P^2=\left[\begin{array}{cc}1& 0\\ 1& 1\end{array}\right]$
$\Rightarrow P^3=\left[\begin{array}{cc}1& 0\\ 1& 1\end{array}\right]\left[\begin{array}{cc}1& 0\\ 1/2& 1\end{array}\right]$
$\therefore P^3=\left[\begin{array}{cc}1& 0\\ 3/2& 1\end{array}\right]$
$\therefore P^n=\left[\begin{array}{cc}1& 0\\ n/2& 1\end{array}\right]$
Now,
$\therefore P^{50}=\left[\begin{array}{cc}1& 0\\ 25& 1\end{array}\right]$