Multiplicatively closed subspaces of M_3(C)

[ $\star \star \star$] Let $V=\mathbb{M}_3(\mathbb{C}),$ the set of all $3 \times 3$ matrices with entries in $\mathbb{C}.$ Clearly $V$ is a vector space over $\mathbb{C}$ and $\dim_{\mathbb{C}} V= 9.$

Let $W$ be a vector subspace of $V$ such that $uv \in W$ for all $u,v \in W.$ Prove that $\dim_{\mathbb{C}} W \neq 8.$