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 $\left[ \begin{array} { l l l }
0.5 & 0 & 0.5
\end{array} \right]$

Question

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 $\left[ \begin{array} { l l l } 
0.5 & 0 & 0.5
\end{array} \right]$

Quizplus · Accepted Answer

The two-step transition matrix is obtained by squaring the original transition matrix $P$, and the distribution vector after two steps is found by multiplying the initial distribution vector $v$ by the two-step transition matrix. The correct two-step transition matrix and distribution vector match those in option C, indicating the correct calculation of both the matrix multiplication for the two-step transition and the subsequent vector-matrix multiplication for the distribution vector.

You Are Given a Transition Matrix P and Initial Distribution P=[0.10.9001000.20.8]P = \left[ \begin{array} { c c c } 0.1 & 0.9 & 0 \\0 & 1 & 0 \\0 & 0.2 & 0.8\end{array} \right]P=​0.100​0.910.2​000.8​​

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