Answer
Answers to a, b, c, d, are all
$\left[\begin{array}{ll}
0.1 & 0.9\\
0.1 & 0.9
\end{array}\right]$
Work Step by Step
$P^{2}=\left[\begin{array}{ll}
0.1 & 0.9\\
0.1 & 0.9
\end{array}\right]\left[\begin{array}{ll}
0.1 & 0.9\\
0.1 & 0.9
\end{array}\right]$
$= \left[\begin{array}{ll}
0.1\cdot 0.1+0.9\cdot 0.1 & 0.1\cdot 0.9+0.9\cdot 0.9\\
0.1\cdot 0.1+0.9\cdot 0.1 & 0.1\cdot 0.9+0.9\cdot 0.
\end{array}\right]=\left[\begin{array}{ll}
0.1 & 0.9\\
0.1 & 0.9
\end{array}\right]=P$
So, $P^{2}=P.$
$b.$
$P^{4}=P^{2}\cdot P^{2}=P\cdot P=P^{2}=P$
$c.$
$P^{8}=P^{4}\cdot P^{4}=P\cdot P=P^{2}=P$
$d.$
All powers of P equal P, so
$P^{1,000}=P$
