Generalized Pell Equations for 2 × 2 Matrices

Abstract

In this paper we consider the solutions of the generalized matrix Pell equations X2 − dY2 = cI, where X and Y are 2 × 2 matrices over ℤ, d is a non-zero (positive or negative) square-free integer, c is an arbitrary integer and I is the 2 × 2 identity matrix. We determine all solutions of such equations for c = ±1, as well as all non-commutative solutions for an arbitrary c.

Keywords

• matrix equations
• pell equation
• 15a24
• 15b36
• 11d09