Let M_q denote the space of q×q matrices, and let P(M_q) denote its projectivization. For a matrix x = (x_ij) we consider two maps. One is J(x) = (x_ij⁻¹) which takes the reciprocal of each entry of the matrix, and the other is the matrix inverse I(x) = (x_ij)⁻¹. The involutions I and J, and thus the mapping K = I ◦ J, arise as basic symmetries in Lattice Statistical Mechanics. This leads to the problem of determining the iterated behavior of K on P(M_q). A basic question is to know the degree complexity δ(K) := lim _n→∞ (deg(Kⁿ))^1/n = lim _n→∞ (deg(K ◦ · · · ◦ K))^1/n of the iterates of this map. (The quantity log δ is also called the algebraic entropy in the paper [BV].)