Find the steady state vector for

We didn't get to go over this section yet it appeared on the online homework. So I'm stuck.

2. Apparently, steady state vector is an eigenvector of the matrix with eigenvalue 1. I.e., if a matrix is $\displaystyle M$, then $\displaystyle (x,y)M=(x,y)$. Therefore, $\displaystyle (x,y)(M-I)=0$, where $\displaystyle I$ is the identity matrix (with 1's on the diagonal and 0 everywhere else).

$\displaystyle M-I=\left( \begin{array}{cc} -0.5 & 0.5\\ 0.8 & -0.8 \end{array} \right)$
So we get an equation $\displaystyle 0.5x-0.8y=0$, or $\displaystyle 5x-8y=0$. Besides, $\displaystyle x$ and $\displaystyle y$ are probabilities, so $\displaystyle x+y=1$.