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Convert the following expression into a determinant?

Posted 5 months ago
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Suppose I have the expression:

c q u - i q x - c w y + e x y - e u z + i w z

I'd like to be able to run FullSimplify or Simplify on it and get this:

Det[{{a, b, c}, {d, e, f}, {g, h, i}}]

One can argue that the evaluated form is actually simpler and hence Mathematica would evaluate the expression but in my case I have a several lines long expression which I suspect could be greatly simplified using matrix notation / determinants and I could use the symbolic horsepower of Mathematica to do the crunching for me. However I'm not sure how to tell it to work "backwords" and try to figure out how to best represent the expression with determinants. So far I've tried Assumptions and custom TransformationFunctions without much luck

For those of you curious about the actual expression, here it is - attaching a sample nb file /its a solution to a linear algebra problem, so there has to be a more elegant matrix representation of this ugly mess/ The variables are 3 3d points: p1, p2 and o each represented by their coords as a list, such as {x1,y1,z1}, and grade which is a real number

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Looks like there are multiple determinants that represent an expression.

In[3]:= Reduce[
 Det[{{a1, b1}, {c1, d1}}] == Det[{{a, b}, {c, d}}], {a1, b1, c1, d1}]

Out[3]= (a1 != 0 && d1 == (-b c + b1 c1 + a d)/a1) || (a1 == 0 && 
   b1 != 0 && c1 == (b c - a d)/b1) || (d == 0 && c == 0 && a1 == 0 &&
    b1 == 0) || (d != 0 && a == (b c)/d && a1 == 0 && 
   b1 == 0) || (d == 0 && c != 0 && b == 0 && a1 == 0 && b1 == 0)
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