  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.
