Determinants and Inverses of Circulant Matrices with Jacobsthal and Jacobsthal-Lucas Numbers
Provides closed-form results for a specific class of number-theoretic circulant matrices, which is an incremental contribution to the study of structured matrices.
The paper derives explicit formulas for the determinants and inverses of circulant matrices whose entries are Jacobsthal and Jacobsthal-Lucas numbers, proving that these matrices are always invertible.
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}.