Given a markov chain where the next state is related to the previous state by the following matrix: $\begin{array}{c|ccc} & A & B & C\\ \hline A & p_1 & q_1 & r_1\\ B & p_2 & q_2 & r_2\\ C & p_3 & q_3 & r_3\\ \end{array}$ Where the system of equations is given by
A$_{n+1} = p_1A_n + p_2B_n+p_3C_n$
B$_{n+1} = q_1A_n + q_2B_n+q_3C_n$
C$_{n+1} = r_1A_n + r_2B_n+r_3C_n$
How can the final relationships be found? (i.e. the limit of each function as n $\rightarrow \infty$, assuming A$_0 = B_0 = C_0$).