I would guess that we are looking to find $\displaystyle \lim_{n \to \infty} P^n S_0$. For example: Solution to part a.
-Dan
Therein lies my confusion. I would guess that we are looking to find:
$$\lim_{n \to \infty} S_0 P^n$$
For example, Solution to part a.
A You were asked about whether your text book uses the convention that applying the matrix, A, to the vector, v, is Av, with v a column matrix, or vA with v as a row matrix. When you say "this is what my text book has" do you mean that your text book says nothing about what a "state matrix" is or how the multiplication is defined? It just has that problem even though nothing has been said about "matrices" before?
Based on the text:
$$Q = \begin{pmatrix}0.2 & 0.2 \\ 0.3 & 0.3\end{pmatrix}$$
$$F = (I-Q)^{-1} = \begin{pmatrix}0.8 & -0.2 \\ -0.3 & 0.7\end{pmatrix}^{-1} = \begin{pmatrix}1.4 & 0.4 \\ 0.6 & 1.6\end{pmatrix}$$
$$FR = \begin{pmatrix}0.36 & 0.64 \\ 0.44 & 0.56\end{pmatrix}$$
This means that:
$$\bar{P} = \begin{pmatrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0.36 & 0.64 & 0 & 0 \\ 0.44 & 0.56 & 0 & 0\end{pmatrix}$$
Now, it is still an open question about if the multiplication is with row vectors or column vectors.