Matrix is a classic walled garden - it can connect to other systems, but provides no means for users of those other systems to connect to it. Matrix users can join IRC channels, but IRC users cannot join Matrix rooms.

And people wonder why I hate Matrix users on IRC, with all their dumb Matrix square-peg-round-hole nonsense messages

@sir there exists a matrix client that presents as an irc server: https://github.com/matrix-org/matrix-ircd . It's not like it's impossible, someone just needs to write the code. And for matrix the spec is open, so it's easier than reverse engineering the many systems that matrix can bridge.