Consider the continuous-time markov chain with state space {1,2,3,4} and infinitesimal generator $$A=\begin{bmatrix}-2&1&1&0\\0&-1&1&0\\1&1&-3&1\\0&0&1&-1\end{bmatrix}$$
a)Find the equilibrium distribution $\pi$
b)Suppose the chain starts in state 1. What the expected amount of time until it changes states for the first time?
c)Again assume the chain starts in state 1. What is the expected amount of time until the chain is in state 4?
a)I solve the system $$\pi A=0$$ with the condition $$\pi(1)+\pi(2)+\pi(3)+\pi(4)=1$$ and get $$\pi=(\frac{1}{8},\frac{3}{8},\frac{2}{8},\frac{2}{8})$$
b)If we start at state $1$ we have that $$a(1,2)=1$$ $$a(1,3)=1$$ $$a(1,4)=0$$ so the change rate is $\sum_j a(1,j)=2 \forall i\neq j$ then the amount of time is $$\frac{1}{\sum_j a(1,j)}=\frac{1}{2}$$
c)Here I am lost, the idea presented in the book involves calculating the inverse matrix of $A$ without the column and row of state $4$