Corolla based on Toyota's web site is (manual tranny) 40/53 (city/highway) miles to the gallon were as the Matrix is 36/48. Of course this is also based on lab testing you may get better or worse depending on how you drive and the type of driving.
I haven't driven a Matrix but was told by someone that owns one that once you do you will want to buy it. If you have the jack and want AWD then the Matrix is the way to go. If you want economy then buy the Corolla.
I will leave it to the rest of the group to get into the detailed specification difference between the 2.