A contribution to the connections between Fibonacci numbers and matrix theory

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 $n\times n$ $\left(0,1\right)$-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.