I am having trouble understanding how to answer this question. I have a transition matrix P: $$ \begin{matrix} .7 & 0 & .3 \\ 0 & 1 & 0 \\ .2 & 0 & .8 \\ \end{matrix} $$ I've figured out the first part of the question which asked me to prove that two 1x3 matrices were stationary which I did with SP=S. But for the second part of the question it asks me to find another stationary matrix for P and it provides me with a hint. The hint is:
$$ \begin{matrix} \\T=aR+(1-a)S, \\ 0<a<1 \end{matrix} $$
I'm not sure what to do with that equation. Am I supposed to turn it into another 1x3 matrix and prove it's stationary with TP=T? Do I multiply the equation as it is now with P and then figure out something with that result? Or do I substitute a number in for 'a' and solve for that and then multiply that result with P?
Call one of your given stationary matrices $S$ and call the other one $R$.
The hint is asking you to calculate a linear combination of the two matrices. So pick any value you like for $a$ - perhaps $a=0.5$? and calculate $T$. This will have the property that it is also a stationary matrix.