Multiplicatively closed subsets of $\mathbb{C}^{n \times n}$ Agreeing with Jason in general, I would still mention that one can describe subalgebras of $2\times 2$ matrices. Any 2-dimensional one has the form $\mathbb C[A]$ for some matrix $A$, and is generated so by almost any its matrix. Next, it sems that any 3-dimensional subalgebra is the set of matrces sharing a common eigenvector.