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it would not be a vehicle battery BUT the battery in the REMOTE, is the first thing I would replace, the remote batteries on FACTORY remotes last about 1 year before they need replacing, as for why the alarm goes off when you unlock the jeep that is because there is NO switch in the door to disarm the alarm, they were eliminated in around 2000 when most all of our cars and trucks came with remote keyless entry.
to replace the batteries it depends on the remote you have if you have a metal key with a remote attached look for a small phillips screw on the part where your key attaches to the key ring remove the screw and open the case halves pullout the battery and run to cvs or walgreens and grab the same battery number USUALLY a CR2013 BUT you have to be sure of the battery in the remote and be sure it is put in the correct way.
if after replacing the battery the remote still will not operate then you will need a new remote/key and it will need to be programmed to your jeep
following up to see if you solved your problem or still need assistance.
sorry to hear that i assume you replaced the batteries and bought it second hand but being a 13 it may still have warranty left so in your case a trip to the dealer MAY work out for you