Nonsingular Solutions of the Matrix Equation and the Centralizer of the Matrix

The equation is called the Yang–Baxter-like matrix equation. We examine this equation for matrices of order 2, assuming that is a nonsingular matrix; moreover, we are only interested in nonsingular solutions. Using a unified rule, one can associate each solution with a matrix that commutes with , in other words, with an element of the centralizer of . No obvious reasons exist for two distinct solutions and to generate one and the same element of . Nonetheless, all the solutions, and there are infinitely many of them, produce one and the same matrix in the centralizer. We give an explanation for this amazing fact.