Maximal Mathieu-Zhao Subspaces of <i>M</i> ₂ ( <i>F</i> ) and <i>M</i> _n ( <i>F</i> )

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 Section 2 of this note we give a complete classification of Mathieu subspaces of [Formula: see text] for arbitrary fields [Formula: see text]. In Section 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 char. [Formula: see text].