Fibonacci and Lucas Sums by Matrix Methods

The Fibonacci sequence {Fn} is defined by the recurrence relation Fn = Fn−1+ Fn−2, for n ≥ 2 with F0 = 0 and F1 = 1. The Lucas sequence {Ln} , considered as a companion to Fibonacci sequence, is defined recursively by Ln = Ln−1 + Ln−2, for n ≥ 2 with L0 = 2 and L1 = 1. It is well known that F−n = (−1)Fn and L−n = (−1)Ln, for every n ∈ Z. For more detailed information see [9],[10]. This paper presents an interesting investigation about some special relations between matrices and Fibonacci, Lucas numbers. This investigation is valuable, since it provides students to use their theoretical knowledge to obtain new Fibonacci and Lucas identities by different methods. So, this paper contributes to Fibonacci and Lucas numbers literature, and encourage many researchers to investigate the properties of such number sequences.