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Sounds like it is the base Matrix - probably not as common as the mid-trim (Matrix XR) but they are much more common than the sportier trim Matrix XRS. Seeing that most prefer to get the bells and whistles for their car - cost difference is minimal between the base and XR trim.
But there are a few cases where less options are more desirable - like with fleet use, courier services, delivery, work vehicle, etc - where they might not fully utilize those options. Should be able to tell from a carfax or similar VIN lookup that will tell you what the car was originally used for.
The 2005+ models did come with engine immobilizers (chipped keys) - but may not be on the base level. I know in the case of the 9th gen Corollas, which were mechanically and mostly electrically similar - the base Corollas did not have the chipped keys.
Burned up clocks are pretty common on the 9th gen Corollas and 1st gen Matrix - most just opt to just go without, as the replacement units are pretty expensive, or try and resurrect their clock with a solder modification (check on Youtube, there is a pretty decent tutorial on there).
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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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