Families of polynomials which obey the Fibonacci recursion relation can be generated by repeated iterations of a 2×2 matrix,Q2, acting on an initial value matrix,R2. One matrix fixes the recursion relation, while the other one distinguishes between the different polynomial families. Each family of polynomials can be considered as a single trajectory of a discrete dynamical system whose dynamics are determined byQ2. The starting point for each trajectory is fixed byR2(x). The forms of these matrices are studied, and some consequences for the properties of the corresponding polynomials are obtained. The main results generalize to the so-calledr-Bonacci polynomials.