What happens if you dump antimatter into a black hole? Let's say you have a 10 solar mass BH and dump 10 solar masses of antimatter into the BH... What would happen?
Would I have a 20 solar mass BH?
Would the BH explode 
Do we even understand what would happen to antimatter in the BH?
 A: 
Let's say you have a 10 solar mass BH and dump 10 solar masses of antimatter into the BH... What would happen?

This depends, like most things in relativity, on where the observer is.
First note that it makes no difference whether it's energy or matter (just a form of energy) inside the event horizon - it will stay inside the event horizon from the point of view of an observer outside the horizon.
As far as we know antimatter is, for the purposes of gravitation, the same as matter.  There is no "anti-gravity" effect if you're thinking that.  So the anti-matter and matter "attract" each other (although that's not quite how general relativity describes it formally).
Also, although we talk about mass resulting in the gravitational field, it's more accurate to say that any form of energy contributes to the gravitational field.  So it's really a case of how much energy is inside the event horizon that defines it's size.  Adding more mass or converting mass to energy makes no difference in this sense.  More energy of any form increases the size of the event horizon (with some details in the case of charged particles and extra angular momentum).
So from the observer outside all that happens is that the event horizon gets bigger by the equivalent effect of 10 solar masses.
If you are an observer traveling with the antimatter, then you see the antimatter pass across the event horizon and continue falling into the black hole.  If it encounters matter it will annihilate some of the antimatter.  However once past the event horizon the energy or particles from that annihilation will not leave the black hole.  It will in principle fall to the singularity at which point it doesn't matter what form of energy it is.

Would I have a 20 solar mass BH ?

Yes-ish.
In practical terms that much mass-energy dumped into a black hole quickly would certainly result in the release of gravitational waves that would "use up" up some of that energy, possibly a substantial part of that mass-energy.

Would the BH explode ?

No.
You can't break these things as far as we know.

Do we even understand what would happen to antimatter in the BH?

Whatever could happen it outside, could happen inside.  The energy equivalent will remain inside and someone on the outside can never know what happened it - that's essentially what an event horizon is : events on one side are not knowable by someone on the other side.
A: As the others have answered, our present theories say that you would end up with a double sized black hole.
One way of looking at it is the following:
Suppose the original matter still exists in the singularity. It probably doesn't, but just suppose that it does. 
And you drop anti-matter on it and it annihilates. What happens during annihilation isn't that the matter and anti-matter just disappears, instead both gets converted into radiation.
And now the important point: That radiation cannot escape the singularity!  It is still stuck. It still has the same mass as the matter and anti-matter it was created from.
From outside the black hole it will be impossible to know if the singularity consists of matter, anti-matter or radiation, the total mass is all that counts.
(This thought experiment is probably invalid in that the singularity doesn't remember what it was made from)
A: We have experimental evidence that antiparticles have positive inertial mass. We have no experimental evidence regarding the gravitational mass of antiparticles, because we can’t find or create enough antimatter to measure its gravity. However, I believe most physicists would be extremely surprised if antimatter had antigravity, as there is no reason why it should, and it would violate the equivalence principle. So the answer is almost certainly that you would simply have a 20 solar mass black hole.
