Answer:
S = [0.2069,0.7931]
Explanation:
Transition Matrix:
![P=\left[\begin{array}{ccc}0.31&0.69\\0.18&0.82\end{array}\right]](https://img.qammunity.org/2021/formulas/mathematics/college/58piq6zr4shaxv6y2qevb4bwyjyqj28p2y.png)
Stationary matrix S for the transition matrix P is obtained by computing powers of the transition matrix P ( k powers ) until all the two rows of transition matrix p are equal or identical.
Transition matrix P raised to the power 2 (at k = 2)
![P^(2) =\left[\begin{array}{ccc}0.2203&0.7797\\0.2034&0.7966\end{array}\right]](https://img.qammunity.org/2021/formulas/mathematics/college/koauvmhjro9lzs7a5guh07x7f7vxzsz5qs.png)
Transition matrix P raised to the power 3 (at k = 3)
![P^(3) =\left[\begin{array}{ccc}0.2203&0.7797\\0.2034&0.7966\end{array}\right] X \left[\begin{array}{ccc}0.31&0.69\\0.18&0.82\end{array}\right]](https://img.qammunity.org/2021/formulas/mathematics/college/cw9u1024wn271molf8z84h3o9zjcj8slvo.png)
![P^(3) =\left[\begin{array}{ccc}0.2086&0.7914\\0.2064&0.7936\end{array}\right]](https://img.qammunity.org/2021/formulas/mathematics/college/gzfrdf8l429divkf993xykzfj47a2ehqiz.png)
Transition matrix P raised to the power 4 (at k = 4)
![P^(4) =\left[\begin{array}{ccc}0.2086&0.7914\\0.2064&0.7936\end{array}\right] X \left[\begin{array}{ccc}0.31&0.69\\0.18&0.82\end{array}\right]](https://img.qammunity.org/2021/formulas/mathematics/college/11x8v9dyvuyy7n3zhjqcyxyi7hlkc2te89.png)
![P^(4) =\left[\begin{array}{ccc}0.2071&0.7929\\0.2068&0.7932\end{array}\right]](https://img.qammunity.org/2021/formulas/mathematics/college/4bgk3qqs4c2oylmuhclyoejqhz65vfpaii.png)
Transition matrix P raised to the power 5 (at k = 5)
![P^(5) =\left[\begin{array}{ccc}0.31&0.69\\0.18&0.82\end{array}\right] X \left[\begin{array}{ccc}0.31&0.69\\0.18&0.82\end{array}\right]X\left[\begin{array}{ccc}0.31&0.69\\0.18&0.82\end{array}\right]X\left[\begin{array}{ccc}0.31&0.69\\0.18&0.82\end{array}\right]X\left[\begin{array}{ccc}0.31&0.69\\0.18&0.82\end{array}\right]]()
![P^(5) =\left[\begin{array}{ccc}0.2071&0.7929\\0.2068&0.7932\end{array}\right] X \left[\begin{array}{ccc}0.31&0.69\\0.18&0.82\end{array}\right]](https://img.qammunity.org/2021/formulas/mathematics/college/vxaoi7d2uzxthg24s37ui6payo2i4er7f9.png)
![P^(5) =\left[\begin{array}{ccc}0.2069&0.7931\\0.2069&0.7931\end{array}\right]](https://img.qammunity.org/2021/formulas/mathematics/college/52r6ep0oexkkfwejrw4zu6o7tc66wk5hvn.png)
P⁵ at k = 5 both the rows identical. Hence the stationary matrix S is:
S = [ 0.2069 , 0.7931 ]