About LU factorization

[salvomic]

Dale,
I found this:
if I reorder a matrix like the "permutation" vector given by LU() (i.e. [1,3,2] for our matrix) LU() gives the same pivots like my (much imperfect try to write a program to calculate pivots: see here: GaussJordan() ).

In our example (first LU gave [1,3,2]...) I get:
M1:= [[2,1,1], [-2,7,2], [4,-6,0]]
LU(M1) -> {[1,2,3], [[1,0,0], [-1,1,0], [2,-1,1]], [[2,1,1], [0,8,3], [0,0,1]]} (Doolittle; pivots in U: 2,8,1)
my program gives: gaussJordan([[2,1,1], [-2,7,2], [4,-6,0]]) -> [[2,8,1], [[2,1,1], [0,8,3], [0,0,1]]] (by the way REF(M1) -> [[1,0.5,0.5], [0,1,0.25], [0,0,1]])

I would like to understand how to get the result presented by Strange (pivots: 2, -8, 1 and so on)...
(EDIT: The example of Gil Strange is intended for LU factorization without swapping of rows)