ON CHARACTERIZATIONS OF SOME LINEAR COMBINATIONS INVOLVING THE MATRICES Q AND R

Abstract

Let Q and R be the well-known matrices associated with Fibonacci and Lucas numbers, and k, m, and n be any integers. It is mainly established all solutions of the matrix equations c(1)Q(n) + c(2)Q(m) = Q(k), c(1)Q(n) + c(2)Q(m) = RQ(k), and c(1)Q(n) + c(2)RQ(m) = Q(k) with unknowns c(1), c(2) is an element of C*. Moreover, using the obtained results, it is presented many identities, some of them are available in the literature, and the others are new, related to the Fibonacci and Lucas numbers.