That's just a question of how you define the word "matter", it doesn't tell you anything about whether stuff with negative mass is possible--most people would probably say photons aren't "matter" either, but they exist. I don't know if there's any evidence for negative mass, though in relativity this should be equivalent to negative energy, and physicists do say the space in between parallel plates in the Casimir effect
has negative energy, although without having a quantum theory of gravity I don't think they can be confident that it would actually behave like hypothetical negative mass would be predicted to behave in the context of the current best theory of gravity, general relativity. But I'm pretty sure there's nothing inherently impossible about the theoretical notion of negative mass in general relativity.
If negative mass existed, it wouldn't behave quite like you might expect from science fiction--if you dropped a negative mass object on Earth, it would fall downward just like a positive mass object. But while positive mass would be "universally attractive" in the sense that both positive-mass objects and negative-mass objects would be attracted to a positive-mass planet, negative mass would be "universally repulsive" in the sense that if you could gather a planet-sized collection of negative mass in one place (which would be hard, since the bits would all repel each other), outside positive masses and outside negative masses would both be repelled from the large negative-mass body.