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