  Let A be an n*n matrix with entries a_ij in the field C. Consider the
following two involutive operations on such matrices: the matrix inversion I: A
-> A^-1 and the element-by-element (or Hadamard) inversion J: a_ij -> a_ij^-1.
We study the algebraic dynamical system generated by iterations of the product
JI. In the case n=3, we give the full explicit solution for this system in
terms of the initial matrix A. In the case n=4, we provide an explicit ansatz
in terms of theta-functions which is full in the sense that it works for a
Zariski open set of initial matrices. This ansatz also generalizes for higher n
where it gives partial solutions.
