Geodesic
Generalized pell equations for 2 × 2 matrices
Discussiones Mathematicae. General Algebra and Applications, Tome 37 (2017) no. 1, pp. 13-30

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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 
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     author = {Cohen, Boaz},
     title = {Generalized pell equations for 2 {\texttimes} 2 matrices},
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Cohen, Boaz. Generalized pell equations for 2 × 2 matrices. Discussiones Mathematicae. General Algebra and Applications, Tome 37 (2017) no. 1, pp. 13-30. http://geodesic.mathdoc.fr/item/DMGAA_2017_37_1_a1/