The following paper - https://dl.acm.org/doi/10.1145/800116.803752
states a lemma (lemma 2) about matrices with variable entries but doesnt prove it
the statement:
If you have a matrix with variable entries (say taking real values) such that no variable appears twice in the same row or column, then there is an assignment of the variables such that the resulting matrix is non singular.
I was thinking about how to prove it: one approach would be to show that the determinant is a non-zero polynomial, but i havent succeed in that (except for small matrices 2x2, 3x3 etc). Any help appreciated!