Welcome, Stephen!
A helpful resource is the thread
Trailer Weights in the Real World. Post #297 links to a spreadsheet of the data. According to that spreadsheet, the average weight of a loaded 13' egg-type trailer is 1700#. The heaviest tipped the scales at over 2600#! The lightest, a supposed 800#
Burro, was either helium filled or a gutted empty shell.
It is possible to keep a 13'er under 1500#, but it means forgoing most options and packing very sparingly.
My recollection is that the Matrix was offered with two different drivetrains and was only rated to tow with the larger engine. Could be wrong- check your owner's manual to be sure, not the internet.
IF your Matrix is tow rated, it will certainly specify trailer
brakes. Not all 13'ers have them. Worst case, it could cost over $1K to add a new
axle with
brakes, a controller, and wire it all up, unless you can do the work yourself.
This thread has a basic introduction to some of the parts and terminology related to towing.
Best wishes getting it all sorted out, and hope a nice egg comes your way!