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Suppose that $z_1,z_2,z_3,$ and $w_1,w_2,w_3$ are two triplets of complex numbers. Show that if $w = T(z)$ is the Möbius transformation such that $T(z_j) = w_j$ for $j = 1,2,3$ then it can be described by the equation:

$$ \det\begin{bmatrix} 1&1&1&1\\ z&z_1&z_2&z_3\\ w&w_1&w_2&w_3\\ zw&z_1w_1&z_2w_2&z_3w_3\\ \end{bmatrix} = 0 $$

I dont understand where the determinant comes from. I am assuming (since we are dealing with triplets of points) that we are supposed to use the cross-ratio, but I have never seen a determinant used together with cross-ratio in complex analysis. Any pointers on how to approach a problem like this? What theory am I missing?

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Expand the determinant along the first column to see that the equation is of the form $$ A + Bz + Cw + Dzw = 0 \iff w = \frac{-A-Bz}{C+Dz} =: S(z) $$ for some complex constants $A, B, C, D$. So the equation defines a Möbius transformation $S$. If $(z, w) = (z_j, w_j)$ for some $j$ then the determinant is zero because two columns of the determinant are equal. It follows that $S(z_j) = w_j$ for $j=1,2,3$, and since Möbius transformations are uniquely determined by their images at three distinct points, $S = T$.

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