I have the following Stochastic Matrix $\begin{pmatrix} 0.9 & 0.02 & 0.08 \\ 1-r & r & 0 \\ 0.5 & 0.1 & 0.4 \end{pmatrix}$
I am attempting to work out $P_{i,j} = P(X_n=j | X_0 = i)$ as $n\rightarrow \infty$ which means I need to use the Limit theorem which is $\lim_{n \rightarrow \infty} p_{i,j}^{(n)}= \pi_j$.
Thus I have the simulatenous equations: \begin{align} \pi_1 & = 0.9 \pi_1 + (1-r)\pi_2 +0.5\pi_3 \\ \pi_2 & = 0.02 \pi_1 + r\pi_2 + 0.1 \pi_3 \\ \pi_3 & = 0.08 \pi_1 +0.4 \pi_3\end{align}
Is my solution found simply by solving these simulatenous equations? I can't tell if what I have done is incorrect or not. Help is appreciated.