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.