 Solution

Represent each point by a nonzero column $v_i$. The source's cross-ratio isRescaling representatives cancels between numerator and denominator. If the projective transformation is represented by $A\in {\mathrm{GL}}_2(\mathbb{C})$, then $\det (A{v}_i,A{v}_j)=\det (A)\det (v_i,v_j)$. Two factors of $\det (A)$ occur in both numerator and denominator and cancel. Hence the cross-ratio is invariant under every projective transformation.