Answer
Answers to a,b,c,d are all $\left[\begin{array}{lll}
0.3 & 0.3 & 0.4\\
0.3 & 0.3 & 0.4\\
0.3 & 0.3 & 0.4
\end{array}\right]$
Work Step by Step
$a.$
$P^{2}=\left[\begin{array}{lll}
0.3 & 0.3 & 0.4\\
0.3 & 0.3 & 0.4\\
0.3 & 0.3 & 0.4
\end{array}\right]$
$=\left[\begin{array}{lll}
0.3\cdot 0.3+0.3\cdot 0.3+0.4\cdot 0.3 & 0.3\cdot 0.3+0.3\cdot 0.3+0.4\cdot 0.3 & 0.3\cdot 0.3+0.3\cdot 0.3+0.4\cdot 0.4\\
0.3\cdot 0.3+0.3\cdot 0.3+0.4\cdot 0.3 & 0.3\cdot 0.3+0.3\cdot 0.3+0.4\cdot 0.3 & 0.3\cdot 0.3+0.3\cdot 0.3+0.4\cdot 0.4\\
0.3\cdot 0.3+0.3\cdot 0.3+0.4\cdot 0.3 & 0.3\cdot 0.3+0.3\cdot 0.3+0.4\cdot 0.3 & 0.3\cdot 0.3+0.3\cdot 0.3+0.4\cdot 0.4
\end{array}\right]$
$=\left[\begin{array}{lll}
0.3 & 0.3 & 0.4\\
0.3 & 0.3 & 0.4\\
0.3 & 0.3 & 0.4
\end{array}\right]$
$P^{2}=P $ so
$P^{3}=P^{2}\cdot P=P\cdot P=P^{2}=P,$
$P^{4}=P^{3}\cdot P=P\cdot P=P^{2}=P$
...
(all powers of P equal P)
$P^{4}=P^{8}=P^{1000}=P$
Answers to a,b,c,d are all $\left[\begin{array}{lll}
0.3 & 0.3 & 0.4\\
0.3 & 0.3 & 0.4\\
0.3 & 0.3 & 0.4
\end{array}\right]$
