Writing $P(\omega_{n}=1, \omega_{k} \neq 1: k=0,1, \ldots, n-1)$ in terms of $P(\omega_{k} \neq 1: k = 0, 1, \ldots, n-1))$ for a finite Markov chain

Suppose we have a Markov chain on a finite state space $$X=\{1, \ldots, r\}$$ with transition matrix $$Q = (q_{ij})$$ and initial distribution $$\mu$$. I believe that we can write

$$P(\omega_{k} \neq 1: k = 0, 1, \ldots, n-1))$$

as

$$\mu'Q'\textbf{1}$$

where $$\mu'=[\mu_{2}, \ldots, \mu_{r}]$$, $$Q'$$ is the minor of $$q_{11}$$ (the $$(r-1) \times (r-1)$$ matrix with the first row and column deleted, and $$\textbf{1}$$ is the vector of all ones. I'm wondering is there a way to write

$$P(\omega_{n}=1, \omega_{k} \neq 1: k=0,1, \ldots, n-1)$$

in connection to what's written above, I have written out a few examples and haven't been able to see a pattern other then you take each of the terms above and multiply by $$q_{k1}$$.

I suppose one way to think of this is to think of $$1$$ as an absorbing state....