I believe whatever will fit on the XR will fit the XRS. You'll throw the
handling off, for sure.

And, I don't know about you, but I get attached to my cars. The XRS is
sort of a rare bird. Sure you wouoldn't really rather take the money for
wheels and tires and just buy a beater for the winter? I can imagine tha
places where a Matrix will rot out, quite similar to the areas of the
older Tercel 4WD wagons. They were horrible for rust! I can see the same
areas on the Matrix rotting.

The other alternative would be to have it oiled with hydraulic oil.

First of all, a "rim" is a part of the wheel.  As far as whether a 15" wheel
will fit your Matrix depends on several factors, including bolt pattern,
width, and offset.  Putting a smaller diameter wheel and tire combination
will cause your speedometer and odometer to turn faster and read too high.
It may save you tickets but will also use up your warranty mileage more
quickly.  As others have mentioned, the vehicle will also handle
differently.

Go to a site like tirerack.com and use their calculator to see what size
wheel and tire combination comes closest to your factory size.