I worked as a professional pet groomer for quite some time and am also certified as such after going to school for it. My advice is to shave the mats yourself asap or use your own money to take it to someone that can. Trying to brush mats that are more than a relatively small tangle is not only very painful but also very dangerous. The lightest pressure on the very strained hair pulls it right out of the follicle, so all you're doing is basically spending quite a while "politely" ripping the cats hair out. And those areas, due to the way the hair was ripped out, are then prone to infection and other serious issues.
Brushing out mats is possible, but like I said, only when they are still relatively small and more of a tangle. Truthfully I'd recommend either taking up caring for the cat yourself since I'm sure it's highly unlikely that you'd want to piss off your boyfriend or his mother by taking away a cat that they obviously don't care about but refuse to get rid of.
There is no reason for an animal to go without grooming long enough for mats to form. Period. It's neglect and if you want to take into account the amount of pain it causes the animal along with health issues, it's also abuse.