I think this is false.

Take the first row $A=(1,-1,1,-1,1,0,0)$.

The circulant matrix is:
$$
\left(\begin{array}{rrrrrrr}
1 & -1 & 1 & -1 & 1 & 0 & 0 \\
0 & 1 & -1 & 1 & -1 & 1 & 0 \\
0 & 0 & 1 & -1 & 1 & -1 & 1 \\
1 & 0 & 0 & 1 & -1 & 1 & -1 \\
-1 & 1 & 0 & 0 & 1 & -1 & 1 \\
1 & -1 & 1 & 0 & 0 & 1 & -1 \\
-1 & 1 & -1 & 1 & 0 & 0 & 1
\end{array}\right)
$$

The determinant is $1$.

Your definition is with columns, so you may need
to transpose.