The inverses of some circulant matrices

We present here necessary and sufficient conditions for the invertibility of some circulant matrices that depend on three parameters and moreover, we explicitly compute the inverse. Our study also encompasses a wide class of circulant symmetric matrices. The techniques we use are related with the solution of boundary value problems associated to second order linear difference equations. Consequently, we reduce the computational cost of the problem. In particular, we recover the inverses of some well known circulant matrices whose coefficients are arithmetic or geometric sequences, Horadam numbers among others. We also characterize when a general symmetric, circulant and tridiagonal matrix is invertible and in this case, we compute explicitly its inverse.