You Are Given a Transition Matrix P and Initial Distribution $P = \left[ \begin{array} { c c } 0 & 1 \\\frac { 1 } { 3 } & \frac { 2 } { 3 }\end{array} \right]$

Multiple Choice

You are given a transition matrix P and initial distribution vector v. Find the two-step transition matrix and the distribution vector after two steps.
$P = \left[ \begin{array} { c c } 0 & 1 \\\frac { 1 } { 3 } & \frac { 2 } { 3 }\end{array} \right]$ , $v = \left[ \begin{array} { l l } \frac { 1 } { 4 } & \frac { 3 } { 4 }\end{array} \right]$

A) The two-step transition matrix is $\left[ \begin{array} { l l } \frac { 1 } { 3 } & \frac { 2 } { 3 } \\\frac { 2 } { 9 } & \frac { 7 } { 9 }\end{array} \right]$ The distribution vector after two steps is $\left[ \begin{array} { l l } \frac { 1 } { 4 } & \frac { 3 } { 4 }\end{array} \right]$
B) The two-step transition matrix is $\left[ \begin{array} { l l } \frac { 1 } { 3 } & \frac { 2 } { 9 } \\\frac { 2 } { 3 } & \frac { 7 } { 9 }\end{array} \right]$
The distribution vector after two steps is $\left[ \begin{array} { l l } 0 & 1\end{array} \right]$
C) The two-step transition matrix is $\left[ \begin{array} { c c } 0 & 1 \\\frac { 1 } { 9 } & \frac { 4 } { 9 }\end{array} \right]$
The distribution vector after two steps is $\left[ \begin{array} { l l } \frac { 1 } { 4 } & \frac { 3 } { 4 }\end{array} \right]$
D) The two-step transition matrix is $\left[ \begin{array} { l l } \frac { 1 } { 3 } & \frac { 2 } { 3 } \\\frac { 2 } { 9 } & \frac { 7 } { 9 }\end{array} \right]$
The distribution vector after two steps is $\left[ \begin{array} { l l } 0 & 1\end{array} \right]$
E) The two-step transition matrix is $\left[ \begin{array} { l l } \frac { 1 } { 3 } & \frac { 2 } { 9 } \\\frac { 2 } { 3 } & \frac { 7 } { 9 }\end{array} \right]$
The distribution vector after two steps is $\left[ \begin{array} { l l } \frac { 1 } { 4 } & \frac { 3 } { 4 }\end{array} \right]$