Let $A$, $B$, $C$ three complex 2x2 matrices such that $$A^2=B^3=I\;(A\neq I\neq B),\quad ABA=B^{-1},\quad AC=CA,\quad BC=CB.$$
Show that $C=rI$ for some $r\in\mathbb{C}$.
I got the minimal polynomials of $A$ and $B$, but I can't go further to find the minimal polynomial of $C$. Any suggestion?
$A$ and $B$ are diagonalisable. As $A$ doesn't commute with $B$ then $A\ne\pm I$ so $A$ has eigenvalues $1$ and $-1$. It follows that the matrices commuting with $A$ must have the form $rI+sA$. But this can only commute with $B$ if $s=0$. The only possible $C$ are of the form $rI$.