Maximal Mathieu–Zhao subspaces of M2(F) and Mn(F)

Let [Formula: see text] be a field. In 2010, the second author introduced the definition of a Mathieu subspace (or Mathieu–Zhao subspace) for associative [Formula: see text]-algebras, but they are not well-understood even for finite-dimensional algebras. In Sec. 2 of this note we give a complete classification of Mathieu subspaces of [Formula: see text] for arbitrary fields [Formula: see text]. In Sec. 3 we give a construction for a family of maximal Mathieu subspaces of the matrix algebras [Formula: see text] [Formula: see text] of codimension [Formula: see text] for [Formula: see text] when [Formula: see text] has characteristic zero or when [Formula: see text].