If I was mixing two different sources in a matrix what would be the effects of this inversion? Highs not hitting high or lows not thumping low? Would this muddy up a matrix?
That is a really big question. You have to understand a couple of things. First that the real issue at hand here is that the polarity reversal is a big deal. It is true that on playback it is not very noticeable when you flip from + to - on a single source. I'm not sure why that is really. I'd love to find out. I'm not at all a good teacher and also only have a limited grasp of the concepts myself. Maybe there is somewhere online to read up on phase and polarity regarding audio. Here are some basics though. Look at this picture:
Notice the length of the "cycle". It has both "compression" and "rarefaction" in one cycle. Compression is the positive information/pressure, and rarefaction is the negative. This example is a sine wave, a perfect tone. In concert recording your getting what is called a complex waveform. It is combined information from all sorts of frequencies mixed together. That is why it doesn't look smooth like this example. But the positive information and negative information still move upward and downward from the zero volt line. When your taking the digital out of the effected units the positive voltage is being recorded as negative and the negative is being recorded as positive. I'm no "real expert" but I think this is a big deal. I think that this is the information that is "essentially" telling your speaker to move either inward or outward. Compression = outward, rarefaction = inward. So obviously reversing these would seem to be pretty critical. I am not certain about this last part regarding speakers. Can anyone chime in?
The REALLY BIG part of our question is "what would be the effects of this inversion". Especially now that I know my room mics have been reversed in SO MANY instances I don't know how to answer that very well. I have never really been sure if room mics should have their phase flipped or not to begin with. I have heard engineers say that they should. I'd like to be able to have some time to experiment someday with a PA and a room. I think that generating a low frequency onstage (not through the console alone, maybe a synth miced from an amp and run direct) would be the way to go. All I know for sure is that sometimes I've needed to flip polarity and other times not.
To try to answer a bit, the effect would vary. The reason that low frequencies are normally effected morso than higher ones is something to do with the fact that the waveforms are large for low frequencies. The problem is that when you have 2 sources out of phase frequencies are cut and boosted depending. If something is copied, perfectly inverted 180 degrees and recombined without latency the sound will actually disappear. It is the exact positive and the exact negative canceling each other out. Like 2 + negative 2 equals 0, but with sound. It is physics. On top of this when you combine an equal positive with another positive it increases (doubles) the volume. Like 2 + 2 equals 4. With the complex waveforms being combined, even without the polarity issues this happens in degrees. This is why mixing audio is not "simple" (of course it's not rocket science either). Frequencies don't always "play nice" with each other. You will be getting some frequencies cutting while others boost. The large low frequency waveforms tend to be more noticeable because they line up easier to either boost or cut depending on the phase relation. I can't put in in words well. Like I said I'm not a good teacher. If there was a chalkboard I might have a better chance of getting my point across. So if any of this makes sense you can see how having the polarity improper would make the problems that much more apt to occur. Play around with reversing polarity and sliding sources by milliseconds to hear the effects for your self. Lots of plug ins allow you to reverse polarity with a click of the mouse. DP has one called "invert phase". I know some Waves plugs have a little + box that you can change to - . That is flipping the polarity. So if you have a session handy in which your making a matrix try flipping those to hear the effect.
That fact that different frequencies cycle at different rates makes combining sources more difficult.
I'm not sure how coherent this came out. Maybe some of that info will help you understand the effects. Maybe someone can correct or add to what I said. I definitely have trouble putting thoughts to words sometimes. Especially when it is something like this. I have a good understanding but not a complete comprehension good enough to REALLY get my point across.