Let there be a continuous time Markov chain with three possible states $C_1, C_2, C_3$, and the rate of going from configuration $C_i$ to $C_j$ be $r_{ij}$. A very simple markov chain could be such that $r_{12}=p$ and $r_{23}=q$, and all the other rates are $0$. Then the matrix representation would be \begin{equation} \frac{d}{dt}P = \left[\begin{array}{l}-p&0&0\\p&-q&0\\0&q&0 \end{array}\right] P \end{equation}
Now can I construct a new Markov chain with two states $C, C_3$; where $C$ is a combination of $C_1$ and $C_2$? Meaning $P(C) = P(C_1) + P(C_2)$. So the new matrix representation would be $\frac{d}{dt}P = M P$, and $M$ is a $2 \times 2$ matrix. Now can I find the elements of M?
(Apologise for the lack of mathematical rigour. I come from a background of physics.)