Group inverses of $\{0,1\}$-triangular matrices and Fibonacci numbers

TL;DR

This paper characterizes the possible sums of entries of the inverse of certain 0-1 triangular matrices and extends the results to singular, group invertible matrices, linking matrix properties with Fibonacci numbers.

Contribution

It introduces a characterization of entry sums for inverses of 0-1 triangular matrices and generalizes to singular, group invertible matrices.

Findings

Sum of inverse entries is between 2-F_{n-1} and 2+F_{n-1}

Characterization applies to matrices with entries from {0,1}

Extension to singular, group invertible matrices

Abstract

A number $s$ is the sum of the entries of the inverse of an $n \times n, (n \geq 3)$ upper triangular matrix with entries from the set $\{0, 1\}$ if and only if $s$ is an integer lying between $2-F_{n-1}$ and $2+F_{n-1}$, where $F_n$ is the $n$th Fibonacci number. A generalization of the sufficient condition above to singular, group invertible matrices is presented.