In my research, my computations are giving rise to the following strange phenomena:
Let $$D=\begin{bmatrix}x_1^p & x_2^p & x_{3}^p\\ x_{1}^q & x_{2}^q & x_{3}^q\\ x_{1}^r & x_{2}^r & x_{3}^r \end{bmatrix}$$ and let $\neg x_i^j$ denote the minor of D that is obtained by deleting the row containing $x_i^j$. What I have found is that $$\neg x_1^p \cdot \neg x_2^q -\neg x_1 ^q \cdot \neg x_2 ^p=x_3^r\cdot\det D$$ and similarly for other choices of ordering $p,q,r$ and choices of $1,2,3$. That is, choosing $i,j\in\{i,j,k\}$ instead of $1,2$ and $a,b\in\{a,b,c\}$ instead of $p,q$ in the above expression, will replace the right hand side with $\pm x_k^r\det D$.
Does anyone know of a general mechanism behind this phenomena? I have carried out the computations explicitly, but they were totally unsuggestive of an intrinsic mechanism.