This is my first post here so I apologize if I format something incorrectly.
So I have a transition matrix P, $$ \begin{matrix} .7 & 0 & .3 \\ 0 & 1 & 0 \\ .2 & 0 & .8 \\ \end{matrix} $$ and I have two stationary matrices R and S respectively,
$$ \begin{matrix} .4 & 0 & .6 \\ \end{matrix} $$ and $$ \begin{matrix} 0 & 1 & 0 \\ \end{matrix} $$
I was able to prove that they are both stationary matrices by showing SP=S and RP=R but the next question asks me to find $\mathbf another$ stationary matrix for P using a provided hint. The hint is $$ \begin{matrix} \\T=aR+(1-a)S, \\ 0<a<1 \end{matrix} $$
I am unsure how to get a stationary matrix from that. Do I just multiply the R matrix by a and then multiply (1-a)S and then add the two resulting matrices together? If I do that then I end up with 1-a in the matrix and that just confuses me more. Sorry if this is supposed to be basic stuff, I always struggle with math :/
The hint is actually pointing at the new stationary matrix. Just prove it really is stationary by verifying $TP = T $.