Exact determinants and inverses of (2,3,3)-Loeplitz and (2,3,3)-Foeplitz matrices

Abstract

This paper presents the exact determinants and the inverses of $n\times n$ (2,3,3)-Loeplitz and (2,3,3)-Foeplitz matrices. First, the $n\times n$ (2,3,3)-Loeplitz and (2,3,3)-Foeplitz matrices are introduced. Next, we calculate the determinants and inverses of them by constructing the transformation matrices. Specifically, the exact inverse of the $n\times n$ (2,3,3)-Loeplitz matrix is sparse and can be denoted by the $n\hbox {th}$, $(n+1)\hbox {st}$ and $(n+2)\hbox {th}$ Fibonacci numbers. The exact determinants of the $n\times n$ (2,3,3)-Foeplitz matrix can be expressed by the $(n+2)\hbox {th}$ Lucas number. The exact inverse of the $n\times n$ (2,3,3)-Foeplitz matrix can be denoted by only seven entries with each entry being the explicit expression of the Lucas or Fibonacci numbers. We also show the exact determinants and inverses of the $n\times n$ (2,3,3)-Lankel and (2,3,3)-Fankel matrices.