A New Method of Matrix Decomposition to Get the Determinants and Inverses of r -Circulant Matrices with Fibonacci and Lucas Numbers

Abstract

We use a new method of matrix decomposition for $r$ -circulant matrix to get the determinants of $A_n={\text{Circ}}_r\left(F_1,F_2,\dots ,F_n\right)$ and $B_n={\text{Circ}}_r\left(L_1,L_2,\dots ,L_n\right)$ , where $F_n$ is the Fibonacci numbers and $L_n$ is the Lucas numbers. Based on these determinants and the nonsingular conditions, inverse matrices are derived. The expressions of the determinants and inverse matrices are represented by Fibonacci and Lucas Numbers. In this study, the formulas of determinants and inverse matrices are much simpler and concise for programming and reduce the computational time.