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Definitely electrical issue here - the problem is where to start.
You mentioned you replaced the door - where did you source the door, and do you still have your original one? What model year Matrix is your car? Factory / aftermarket alarm or keyless entry system?
Could be anything from a pinched/crushed wire from the installation, to a faulty switch in the replacement door.
Have you tried to reset the ECM / factory keyless system to see if it is a body logic issue, sometimes a repair like this will cause all sorts of weird stuff happening. Resetting the ECM / keyless system "fixes" many of these issues. Sort of like rebooting a computer.
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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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