Additionally, if you restrict to invertible matrices with matrix multiplication, they are also a monoid (so they are a submonoid), and in fact a group, the gene

Additionally, if you restrict to invertible matrices with matrix multiplication, they are also a monoid (so they are a submonoid), and in fact a group, the general linear group. If you restrict again to matrices with determinant 1, it's again itself a group (a subgroup, and also a submonoid), the special linear group.