Vol. 8, No. 3, 2015

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ISSN: 1944-4184 (e-only)
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A contribution to the connections between Fibonacci numbers and matrix theory

Miriam Farber and Abraham Berman

Vol. 8 (2015), No. 3, 491–501
Abstract

We present a lovely connection between the Fibonacci numbers and the sums of inverses of (0,1)-triangular matrices, namely, a number S is the sum of the entries of the inverse of an n × n (0,1)-triangular matrix (for n 3) if and only if S is an integer between 2 Fn1 and 2 + Fn1. Corollaries include Fibonacci identities and a Fibonacci-type result on determinants of a special family of (1,2)-matrices.

Keywords
Fibonacci numbers, Hessenberg matrix, sum of entries
Mathematical Subject Classification 2010
Primary: 15A15, 11B39, 15A09, 15B99
Milestones
Received: 19 August 2013
Revised: 28 October 2013
Accepted: 5 November 2013
Published: 5 June 2015

Communicated by Robert J. Plemmons
Authors
Miriam Farber
Department of Mathematics
Technion–Israel Institute of Technology
Technion City, Haifa 3200003
Israel
Abraham Berman
Department of Mathematics
Technion–Israel Institute of Technology
Technion City, Haifa 3200003
Israel