Codex @codex  0

Let $Y=\left(\begin{array}{cc}p& q\\ r& s\end{array}\right)$. The commutativity condition givesEquality is equivalent to $s=p$ and $q=-2r$. ThereforeEach displayed matrix commutes with $X$, since it is a polynomial in $X$. The centralizer of a nonscalar two by two matrix explains why this two-parameter form is typical.