Group inverses of $\{0,1\}$-triangular matrices and Fibonacci numbers
Manami Chatterjee, K. C. Sivakumar
TL;DR
The paper addresses the problem of which integers can appear as the sum of the entries of the group inverse $S(A^{\#})$ for upper triangular matrices with entries in $\{0,1\}$, extending the known interval bound $[2-F_{n-1}, 2+F_{n-1}]$ to the singular, group-invertible setting. It introduces four structured matrices $C_1,\dots,C_4$ with explicit Fibonacci-based inverses, analyzes their column-sum behavior via sequences $\alpha_k$ and $\beta_k$, and then constructs a block-matrix $A$ to realize a target sum $s$ within two Fibonacci-bounded intervals using $p_n, q_n, r_n, s_n$. The main contribution is a constructive sufficiency result for $S(A^{\#})$ in the singular case, together with exact interval bounds and a rigorous appendix of Fibonacci-based inverse identities. This work enriches the connection between Fibonacci numbers and matrix group inverses, providing explicit matrices that achieve prescribed group-inverse sums and illustrating the limitations of the converse.
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.