Invertibility of circulant matrices of arbitrary size

In this paper, we present sufficient conditions to guarantee the invertibility of rational circulant matrices with any given size. These sufficient conditions consist of linear combinations of the entries in the first row with integer coefficients. Our result is general enough to show the invertibility of circulant matrices with any size and arrangement of entries. For example, using these conditions, we show the invertibility of the family of circulant matrices with particular forms of integers generated by a primitive element in ${\mathbb{Z}}_p$. Also, using a combinatorial structure of these sufficient conditions, we show invertibility for circulant $0,1$-matrices.