My long-haired tortoise shell had that problem about a year and a half ago. I did opt to cut the mats out a little at a time. The vet gave me some sedatives for her but that only slowed her down - didn't put her out. Did managed to get a little bit of it done that day. Each day after that we had a routine of brushing her and then getting the scissors to cut the mats out. It took about a week to get them all out. You must be very careful but it can be done. I figured the risk of cutting them out outweighed the risk of having her put out with anesthesia. It worked for me but I wouldn't recommend it for everyone. One must have a very steady hand and even then it's risky to cut them out, especially on hind quarters where one slip of the scissors could be disastrous. Since then, my feline, who thinks she's a goddess, gets a healthy brushing every day and she has grown to absolutely love it. We bond, she purrs, the job gets done, we're all happy.

Good luck!