## Abstract

The algebraic structure of the set of all Fibonacci-like sequences, which includes the Fibonacci and Lucas sequences, is developed, utilizing an isomorphism between this set and a subset of the 2 by 2 integer matrices. Using this isomorphism, determinants of sequences, and Fibonacci-like matrices, can be defined. The following results are then obtained: (1) the Fibonacci sequence is the only such sequence with determinant equal to 1, (2) the set of all Fibonacci-like sequences forms an integral domain, (3) even powers of Lucas matrices are multiples of a Fibonacci matrix, and (4) only powers of multiples of Fibonacci matrices or Lucas matrices are multiples of Fibonacci matrices.

## Faculty Sponsor

Charlie Jacobson

## Recommended Citation

Parry, Stephen
(2008)
"Unique Properties of the Fibonacci and Lucas Sequences,"
*Rose-Hulman Undergraduate Mathematics Journal*: Vol. 9:
Iss.
1, Article 5.

Available at:
https://scholar.rose-hulman.edu/rhumj/vol9/iss1/5