It looks like the immobilizer is a separate ECU according to my finding on Google. But, it would really help if we know what the key looks like.
Edit : whomever you dealt with at the dealer is squeezing a heck of money from you. It's very apparent you don't need the immobilizer ECU. My findings show it's a separate computer. Here's what a dealer will charge for the ECU. Note, I selected 2009 matrix 1.8 liter engine with automatic transmission.
Yes tried it. I recently bought a. Used ecu with identical numbers from eBay. I’ll know by the end of the week if it works. My locksmith wants to copy the program if it does. He is scratching his head trying to fix this issue also. He is the guy the dealership calls when they can’t fix it.