  When $\mathbb{K}$ is a field, and $\mathcal{A}$ and $\mathcal{B}$ denote
commuting subspaces of $\text{M}_n(\K)$ each of which contains a non-scalar
matrix, we prove that $\dim \mathcal{A} +\dim \mathcal{B} \leq (n-1)^2+3$. We
also give a complete description of the cases when equality holds.
