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I believe ABS was an option for the XR trim, so unless you see an ABS module under the hood or see an ABS warning lamp on the dash when you cycle the key to the ON position.
Since you mentioned braking on wet surfaces is poor - how does the car brake on dry? If it seems to perform better - two possible things are happening. One, it is your tires that are the source of the problem. Doesn't matter how strong the brakes are - if the tires cannot produce enough traction, your braking performance will be poor. Second, could be getting water onto the rotors/drums - only if you drive through some pounding pain or ford through some pools of water. That case, you need to pump the brakes to clear off that water and dry the brakes out to maximize their effectiveness.
You're also trying to compare a BMW to a Matrix - not even remotely close comparison. BMW is a RWD platform with close to 50/50 weight split, relatively big brakes and meaty performance tires. Matrix is FWD built on a economy car platform (Corolla) with a 60/40 weight split, Corolla brakes and tires.
If the performance is the same, wet or dry - could be either the pads are not up to par (glazed, overheated, not bedding into the rotor or not compatible with the braking system (some pads require newly turned rotors or they will not bed in properly), could also be a braking system issue (rotors are too thin, not enough heat capacity - so brakes will overheat), same with brake fluid - should be changed regularly to maximize performance.
Upgrading to a performance street pad - like Hawk or Porterfield - can help increase brake torque and initial bite so that it "feels" like the brakes are more powerful. Going with a high performance street pad (not a race pad) will generally retain good cold weather brake performance, and still be easy on the rotors. Some of the best modifications you can do to this car are running better tires (OEM tires are not that great) and upgrading to a higher performance street pad.
For the rear brake shoes - OEM would be my recommendation - since the car has a heavy front weight bias, and just from the normal weight transfer during braking - the rear brakes only do a fraction of the braking. You can upgrade to rear disc brakes will not yield a quantum leap in braking performance - but they can help some. Unless you have a system to help transfer some of that load - i.e., Toyota's EBD system - which helps level the car by modulating how much and when the brakes "grab" - part of their ABS system on the Matrix XRS. Probably not something you can easily mod into the Matrix XR, atleast without spending a considerable amount of money (could be looking at several thousands of dollars).
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2002 Corolla S, 1.8L 1ZZ-FE VVT-i
2003 Matrix XRS, 1.8L 2ZZ-GE, VVTL-i (RIP)
2009 Matrix XRS, 2.4L 2AZ-FE VVT-i
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