My oldest cat gets mats in her fur too. She's a short hair cat, but not as short hair as some cats I've seen.
I find that if I don't brush her daily she will have mats in her fur at the lower end of her back near her tail. There was a period where I was quite sick and it was all I could do to feed them and keep their litter box cleaned, so they didn't get groomed for several months. Chynna (the oldest) had knots and her fur looked greasy in the area.
Holding the knot at the base next to the skin, I gently worked the knot with my fingers and kind of spread it apart a bit and brushed with my wire slicker brush at the very end of the knot, while still holding it tight against the skin. It only took me 3 days to get all of her mats out and I didn't have to cut any of her fur off either.
I found that each time I worked at the knot and spread it apart as I could and then brushed it gently, that the matted dead hair gradually came out.
However, if your cat doesn't like to be brushed, that may not work for you. Chynna loves to be brushed and will lay there all day and let me brush her.
I also found that once I got the knots out and brushed her daily that the greasy feel and look of her fur disappeared and her fur returned to feeling and looking silky soft like goose down.