On the relations of Fibonacci to the determinants of the sum of the Fibbonacci-Hessenberg matrices

The Fibonacci sequence is de ined by fo  O f, 11 and fn    fn 1                                     fn    2;n  2.

An n n matrix A is called a (lower) Hessenberg matrix if all entries above the superdiagonal are zero. A Fibonacci-Hessenberg matrices are matrices whose determines are in the form tfn _1 fn₂orfn₁tfn₂ for some real or complex number t . De ine n n and an n n matrix n t_{, ,} a new matrix is formed by adding one row of weight one and starting with alternating 1’s and 0’s, starting a 1, to the left of the obtained matrix. Let i denote the usual complex unit with i² 1. Replacing the entry of n t_, located in the i^th row (i₁₎^th  column, 1 i n, with ( 1)_ ^{i n} i , we obtain another Hessenberg matrix denoted H_{n t, .} In this paper, we take the sum of _{n t,} and H_{n t,} and show that the determinants of the sum of the two hessenberg matrices are not equal to the sum of the determinants of the two hessenberg matrices. We also provide illustrations and ibonacci relations of the determinants of the resulted sum.