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.
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).
In your case,
$\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 $\displaystyle x+y=1$.
However, if you have not covered this topic, it is probably instructor's oversight. It's possible that he/she prepared the plan with homeworks in advance but has not had time to cover this in class. It's better to contact your instructor and ask what to do.
Another thing is, this is not really a problem for this subforum. Since this is about Markov chains, you should post it to something like probability theory section.