On the determinants and inverses of circulant matrices with Fibonacci and Lucas numbers

Let An=Circ(F1,F2,…,Fn)An=Circ(F1,F2,…,Fn) and Bn=Circ(L1,L2,…,Ln)Bn=Circ(L1,L2,…,Ln) be circulant matrices, where Fn is the Fibonacci number and Ln is the Lucas number. We prove that AnAn is invertible for n > 2, and BnBn is invertible for any positive integer n  . Afterwards, the values of the determinants of matrices AnAn and BnBn can be expressed by utilizing only the Fibonacci and Lucas numbers. In addition, the inverses of matrices AnAn and BnBn are derived.