Vol. 8, No. 3, 2015

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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 $\left(0,1\right)$-triangular matrices, namely, a number $S$ is the sum of the entries of the inverse of an $n×n$ $\left(0,1\right)$-triangular matrix (for $n\ge 3$) if and only if $S$ is an integer between $2-{F}_{n-1}$ and $2+{F}_{n-1}$. Corollaries include Fibonacci identities and a Fibonacci-type result on determinants of a special family of $\left(1,2\right)$-matrices.

Keywords
Fibonacci numbers, Hessenberg matrix, sum of entries
Mathematical Subject Classification 2010
Primary: 15A15, 11B39, 15A09, 15B99