A contribution to the connections between Fibonacci Numbers and Matrix Theory

TL;DR

This paper explores a novel connection between Fibonacci numbers and matrix theory, specifically relating sums of inverses of (0,1)-triangular matrices to Fibonacci bounds, leading to new identities and determinant results.

Contribution

It establishes a new relationship between Fibonacci numbers and the sums of inverses of (0,1)-triangular matrices, including deriving Fibonacci identities and determinant properties.

Findings

Sum of inverses is an integer within Fibonacci-based bounds.

Derived new Fibonacci identities from matrix properties.

Established Fibonacci-like results for determinants of (1,2)-matrices.

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 $n \times n$ $(n \geq 3)$ $(0,1)-$ triangular matrix iff $S$ 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 family of (1,2)-matrices.