  We consider two (densely defined) involutions on the space of $q\times q$
matrices; $I(x_{ij})$ is the matrix inverse of $(x_{ij})$, and $J(x_{ij})$ is
the matrix whose $ij$th entry is the reciprocal $x_{ij}^{-1}$. Let $K=I\circ
J$. The set ${\cal SC}_q$ of symmetric, cyclic matrices is invariant under $K$.
In this paper, we determine the degrees of the iterates $K^n=K\circ...\circ K$
restricted to ${\cal SC}_q$.
