@Bohdan I have a very low personal psychological tolerance for GUI sluggishness, and your worksheet "Problem_3.mw" for the12x12 system slightly exceeds that tolerance. Would it be possible for us to "practice" similar simplification operations with a smaller system, say the 8x8 system that you referred to earlier (if it's full rank)? There are easy and very fast commands that can replace all the operations that you laboriously do with the mouse. One such command is numer, which in this case will extract the numerator polynomial of a rational function. (I think that you realize already that all the denominator polynomials are the same, being the determinant of the coefficient matrix of the original system.) I'd be happy to teach you these commands on a smaller system.
Is this statement correct: No term in any of the expanded numerators contains more than one rho?
My guess is that you want to express these numerators in the form
Sum(rho[i,r]*Sum(Product(pi[...], ...)*Product(gamma[...], ...), ...)
Is that correct?
The hard part is seeing the patterns of the indices. I don't think that simplify will help you see that. What might help, though, is having the terms sorted by those indices. Do you agree? This can be easily done.