Combinatorial Identities Deriving from the n-th Power of a 2 X 2 Matrix

Authors

James McLaughlin, West Chester University of PennsylvaniaFollow

Document Type

Article

Publication Date

2004

Abstract

In this paper we give a new formula for the n-th power of a 2 × 2 matrix. More precisely, we prove the following: Let A = (a b c d) be an arbitrary 2 × 2 matrix, T = a + d its trace, D = ad − bc its determinant and define yn : = b X n/2c i=0 (n − i i )T n−2i (−D) i . Then, for n ≥ 1, A n = (yn − d yn−1 b yn−1 c yn−1 yn − a yn−1) . We use this formula together with an existing formula for the n-th power of a matrix, various matrix identities, formulae for the n-th power of particular matrices, etc, to derive various combinatorial identities.

Publication Title

INTEGERS: The Electronic Journal of Combinatorial Number Theory

ISSN

1553-1732

Publisher

Colgate University, Charles University, and DIMATIA

Volume

4

Issue

A19

First Page

1

Last Page

14

Comments

Preprint version is available here.

Recommended Citation

McLaughlin, J. (2004). Combinatorial Identities Deriving from the n-th Power of a 2 X 2 Matrix. INTEGERS: The Electronic Journal of Combinatorial Number Theory, 4(A19), 1-14. Retrieved from https://digitalcommons.wcupa.edu/math_facpub/39