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

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