Az Eszterházy Károly Tanárképző Főiskola Tudományos Közleményei. 1995-1996. Sectio Mathematicae. (Acta Academiae Paedagogicae Agriensis : Nova series ; Tom. 23)

GRYTCZUK, A. and VOROBEV, N. T., On some applications of 2 X 2 integral matrices

On some applications of 2 x 2 integral matrices A. GRYTCZUK and N. T. VOROBÉV Abstract. In this paper we give a matrix representation for the fundamental solution of the Pellian type equation x

2-dy

2 — -l. Using matrices the solutions of linear equations are also represented. In 1970, in [1] some connections was given between integral 2x2 matrices and the Diophantine equation ax — by = c. Namely, we proved that the solution {Xo,yo

) of this equation can be determined by the following equalities: <•) C S)-(S !)"(! !)•-(! !)'"(? T) if m is even, and « (::)-(i!)"(i ?)••••(:!)'(!:) if M is odd, where | = [Q0',QI , • • •, Qm] is a representation of | as a simple finite continued fraction. For example, consider the equation 19x - 11 y = —2. We have = [1; 1,2,1,2] and consequently q

0 = 1 ,q\ = 1,^2 = = 1, g4 = 2, thus m — 4 and by (1) we obtain »

(s :)•(! DC f)(s:)"(! 00:)'(! :)• By Cauchy's theorem on product of determinants it follows from (3) that (4) 19x

0 - llyo = -2. So denote that (xq, yo) is an integer solution of the equation 19x — 11 y — —2.