On the sum of the dimension of a matrix subalgebra and its centralizer

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