Let the transitional matrix be $$\begin{pmatrix}0 & 1/2 & 1/2\\\ 1/4 & 0 & 3/4 \\\ 2/3 & 1/3 & 0\end{pmatrix}$$ The matrix rows and columns represent point $1, 2$ and $3$ in order. Find the probability that, given you start in Turn $0$ at $1$, that you are at $0$ after $n$ steps? I know of the diagonalising method for this, but not how to do the method using difference equations.
Can someone give me the method for this please?