Summary: Let \(\mathcal{A}\) be an \(n\times n\) matrix with entries \(a_{ij}\) in the field \(\mathbb{C}\). We consider two involutive operations on these matrices: the matrix inverse \(I: {\mathcal A}\mapsto{\mathcal A}^{-1}\) and the entry-wise or Hadamard inversion \(J: a_{ij}\mapsto a_{ij}^{-1}\). We study an algebraic dynamical system generated by iterations of the product \(J\circ I\). We construct the complete solution of this system for \(n\leq 4\). For \(n = 4\), it is obtained using an ansatz in theta functions. For \(n\geq 5\), the same ansatz gives partial solutions. They are described by integer linear transformations of the product of two identical complex tori. As a result, we obtain a dynamical system with mixing described by explicit formulas.