Not clear but possibly you want to triangularize the matrix withpout normalizing pivot positions? If so then the Wolfram Function Repository function to triangularize a matrix will handle this, For input matrix mat it returns an upper triangular matrix red and a conversion matrix mult such that mult . mat = red.

mat = {{2, -1, -1}, {-1, 2, -1}, {-1, -1, 2 + c}};
{red, mult} = ResourceFunction["UpperTriangularDecomposition"][mat]

(* {{{-1, 2, -1}, {0, -3, 3}, {0, 0, -3 c}}, {{0, 1, 0}, {-1, -2, 0}, {-3, -3, -3}}} *)

Check:

mult . mat - red // Expand

(* Out[111]= {{0, 0, 0}, {0, 0, 0}, {0, 0, 0}} *)

The symbolic pivot in this case is -3 c.