You Are Given a Transition Matrix P and Initial Distribution $P = \left[ \begin{array} { c c c } 0.1 & 0.9 & 0 \\0 & 1 & 0 \\0 & 0.2 & 0.8\end{array} \right]$

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.
$P = \left[ \begin{array} { c c c } 0.1 & 0.9 & 0 \\0 & 1 & 0 \\0 & 0.2 & 0.8\end{array} \right]$ , $v = \left[ \begin{array} { l l l } 0.5 & 0 & 0.5\end{array} \right]$

A) The two-step transition matrix is $\left[ \begin{array} { c c c } 0.01 & 0.99 & 0 \\0 & 1 & 0 \\0 & 0.64 & 0.36\end{array} \right]$ The distribution vector after two steps is $\left[ \begin{array} { l l l } 0.005 & 0.675 & 0.32\end{array} \right]$
B) The two-step transition matrix is $\left[ \begin{array} { c c c } 0.01 & 0.99 & 0 \\0 & 1 & 0 \\0 & 0.36 & 0.64\end{array} \right]$
The distribution vector after two steps is $\left[ \begin{array} { l l l } 0 & 1 & 0\end{array} \right]$
C) The two-step transition matrix is $\left[ \begin{array} { c c c } 0.01 & 0.99 & 0 \\0 & 1 & 0 \\0 & 0.36 & 0.64\end{array} \right]$
The distribution vector after two steps is $\left[ \begin{array} { l l l } 0.005 & 0.675 & 0.32\end{array} \right]$
D) The two-step transition matrix is $\left[ \begin{array} { c c c } 0.1 & 0.9 & 0 \\0 & 1 & 0 \\0 & 0.36 & 0.64\end{array} \right]$
The distribution vector after two steps is $\left[ \begin{array} { l l l } 0.005 & 0.675 & 0.32\end{array} \right]$
E) The two-step transition matrix is $\left[ \begin{array} { c c c } 0.01 & 0.99 & 0 \\0 & 1 & 0 \\0 & 0.36 & 0.64\end{array} \right]$
The distribution vector after two steps is $\left[ \begin{array} { l l l } 0.5 & 0 & 0.5\end{array} \right]$