Answer
(a)
$P^2=\begin{bmatrix}
.25 & .75 \\
0 & 1 \\ \end{bmatrix}$
(b)
After one step:
$vP=\begin{bmatrix}
.5 & .5 \\ \end{bmatrix}$
After two steps:
$vP^2=\begin{bmatrix}
.25 & .75 \\ \end{bmatrix}$
After three steps:
$vP^3=\begin{bmatrix}
.125 & .875 \\ \end{bmatrix}$
Work Step by Step
(a)
$P^2=\begin{bmatrix}
.5 & .5 \\
0 & 1 \\ \end{bmatrix}\begin{bmatrix}
.5 & .5 \\
0 & 1 \\ \end{bmatrix}=\begin{bmatrix}
.25+0 & .25+.5 \\
0+0 & 0+1 \\ \end{bmatrix}=\begin{bmatrix}
.25 & .75 \\
0 & 1 \\ \end{bmatrix}$
(b)
$P^3=P^2P=\begin{bmatrix}
.25 & .75 \\
0 & 1 \\ \end{bmatrix}\begin{bmatrix}
.5 & .5 \\
0 & 1 \\ \end{bmatrix}=\begin{bmatrix}
.125+0 & .125+.75 \\
0+0 & 0+1 \\ \end{bmatrix}=\begin{bmatrix}
.125 & .875 \\
0 & 1 \\ \end{bmatrix}$
After one step:
$vP=\begin{bmatrix}
1 & 0 \\ \end{bmatrix}\begin{bmatrix}
.5 & .5 \\
0 & 1 \\ \end{bmatrix}=\begin{bmatrix}
.5 & .5 \\ \end{bmatrix}$
After two steps:
$vP^2=\begin{bmatrix}
1 & 0 \\ \end{bmatrix}\begin{bmatrix}
.25 & .75 \\
0 & 1 \\ \end{bmatrix}=\begin{bmatrix}
.25 & .75 \\ \end{bmatrix}$
After three steps:
$vP^3=\begin{bmatrix}
1 & 0 \\ \end{bmatrix}\begin{bmatrix}
.125 & .875 \\
0 & 1 \\ \end{bmatrix}=\begin{bmatrix}
.125 & .875 \\ \end{bmatrix}$
