Oh, well, I mean that if you have a 3×3 matrix and a 4×4 matrix, you can't even multiply them. So 4×4 matrices form a monoid under multiplication, as do 3×3 matrices, but matrices in general do not.

It turns out that they actually form a much richer structure called a ring when you add in their addition operator as well: https://en.m.wikipedia.org/wiki/Matrix_ring