Determinants and Inverses of Circulant Matrices with Jacobsthal and Jacobsthal-Lucas Numbers

TL;DR

This paper derives formulas for the determinants and inverses of circulant matrices constructed from Jacobsthal and Jacobsthal-Lucas numbers, establishing their invertibility and providing explicit inverse matrices.

Contribution

It introduces explicit formulas for determinants and inverses of circulant matrices based on Jacobsthal and Jacobsthal-Lucas numbers, extending matrix theory with special number sequences.

Findings

Determinants expressed in terms of Jacobsthal and Jacobsthal-Lucas numbers.

Matrices are shown to be invertible.

Explicit formulas for the inverses are derived.

Abstract

Let n\geq3 and J_{n}:=circ(J_{1},J_{2},...,J_{n}) and j_{n}:=\circ(j_{0},j_{1},...,j_{n-1}) be the n\timesn circulant matrices, associated with the nth Jacobsthal number J_{n} and the nth Jacobsthal-Lucas number j_{n}, respectively. The determinants of J_{n} and j_{n} are obtained in terms of the Jacobsthal and Jacobsthal-Lucas numbers. These imply that J_{n} and j_{n} are invertible. We also derive the inverses of J_{n} and j_{n}.