The cube roots of 27 are 3,3ω,3ω2, where ω is a cube root of unity. A circulant matrix like this has a determinant equal to: (x+y+z)(x+ωy+ω2z)(x+ω2y+ωz) For cube roots of unity, one linear factor always becomes zero. Cube roots of 27: x=3,‌‌y=3ω,‌‌z=3ω2 Sum of cube roots of unity is zero: 1+ω+ω2=0 Thus, x+y+z=3(1+ω+ω2)=0 Since one factor in the determinant formula becomes zero: det=(x+y+z)(⋯)(⋯)=0