  Using Wederburn's main theorem and a result of Gerstenhaber we prove that,
over a field of characteristic zero, the maximal dimension of a proper unital
subalgebra in the $n \times n$ matrix algebra is $n^2 - n + 1$ and furthermore
this upper bound is attained for the so-called parabolic subalgebras. We also
investigate the corresponding notion of parabolic coideals for matrix
coalgebras and prove that the minimal dimension of a non-zero coideal of the
matrix coalgebra ${\mathcal M}^n (k)$ is $n-1$.
