Consider the subalgebra of $M_{2n}(k)$ spanned by the matrices of the form $\left(\begin{smallmatrix}0&A\\0&0\end{smallmatrix}\right)$ (all blocks are $n\times n$) together with the identity, which is commutative. Its dimension is larger than $n$, 2n$when$n$is sufficiently large, so it is not generated by a single matrix. 1 Consider the subalgebra of$M_{2n}(k)$spanned by the matrices of the form$\left(\begin{smallmatrix}0&A\\0&0\end{smallmatrix}\right)$(all blocks are$n\times n$) together with the identity, which is commutative. Its dimension is larger than$n\$, so it is not generated by a single matrix.