3-point DFT with two multiplications
What you have to show is that
$$W_3^2+W_3+1=0\tag{1}$$
with $W_3=e^{-j2\pi/3}$. It's a straightforward exercise to prove $(1)$.
In general we have
$$\sum_{k=0}^{N-1}W_N^k=0,\qquad N&gt;1\tag{2}$$
with $W_N=e^{-j2\pi/N}$. Eq. $(2)$ can be shown by using the formula for the geometric sum.