I have to find the steady state vector for matrix is M= [.5 .8
.5 .2]
Would the vector [.8
.5]
be in the steady state because [.5 .8
.5 .2] times x = x so the system reduces down to -.5x+.8y=0 ?
Yes, $\displaystyle \begin{bmatrix}.5 & .8 \\ .5 & .2\end{bmatrix}\begin{bmatrix}.8 \\ .5\end{bmatrix}= \begin{bmatrix}.4+ .4 \\ .4+ .1\end{bmatrix}= \begin{bmatrix}.8 \\ .5\end{bmatrix}$
So that is the "steady state" vector.