Continuous-time Markov Chains transition rate

Question

Why is it that in continuous-time Markov chains we usually have

$$\sum_{j \neq i} q_{ij}(t) = -q_{ii}(t)$$

or alternatively, the transition rate corresponding to the system remaining in place is defined by the equation

$$q_{ii}(t) = -\sum_{j \neq i} q_{ij}(t).$$

Is this always true?

Answer 1

The sum of all the $q_{ij}$ with fixed $i$ is the rate of change of the probability to be anywhere started from $i$, which must be zero since this probability is 1 at all times.