I am not sure I completely understand the definition of (mutual) capacitance. Let's say we have two conductors, $A$ and $B$, so that the following holds:

1. Both conductors are isolated.
2. $A$ is isolated and $B$ is kept at fixed potential.
3. Both conductors are kept at fixed potential.

What is the correct definition (and meaning) of (mutual) capacitance in these three cases? For example, in the first case, we could have arbitrary charges $Q_A$ and $Q_B (\neq - Q_A)$ on the conductors. These charges would result in the potentials $V_A$ and $V_B$. Now, do we define capacitance as $$C = \frac{\Delta Q}{\Delta V},$$ where $\Delta V = V_A - V_B$ and $\Delta Q$ is the charge which would be transferred from $A$ to $B$ if we connected the conductors by a thin wire? If this is correct, the definition still makes sense in the second example but completely fails in the third one. Does it even makes sense to define the capacitance for the system in the third example?

Also, is there a easy way to determine the mutual capacitance from capacitance matrix?

I am not sure I completely understand the definition of mutual capacitance. Let's say we have two conductors, $A$ and $B$, so that the following holds:

1. Both conductors are isolated.
2. $A$ is isolated and $B$ is kept at fixed potential.
3. Both conductors are kept at fixed potential.

What is the correct definition (and meaning) of (mutual) capacitance in these three cases? For example, in the first case, we could have arbitrary charges $Q_A$ and $Q_B (\neq - Q_A)$ on the conductors. These charges would result in the potentials $V_A$ and $V_B$. Now, do we define capacitance as $$C = \frac{\Delta Q}{\Delta V},$$ where $\Delta V = V_A - V_B$ and $\Delta Q$ is the charge which would be transferred from $A$ to $B$ if we connected the conductors by a thin wire? If this is correct, the definition still makes sense in the second example but completely fails in the third one. Does it even makes sense to define the capacitance for the system in the third example?

Also, is there a easy way to determine the mutual capacitance from capacitance matrix?