Interchange row 1 and row 3 [left multiplication by P

Do elimination on the first column [multiplication by L

Interchange rows two and four [multiplication by P

Elimination on the second column [multiplication by L

Interchange rows three and four [multiplication by P

Elimination on the third column [multiplication by L

Collecting the pieces we have

where U is the upper triangular matrix on the right just above -- but the factors to the left of A are not at all lower triangular. We are lucky there is a fairly simple way out. These six elementary operations can be reordered in the form

where L'

Since each of these definitions apply only permutations P

as in equation (*).

The product of the matrices L'_{k} is also unit lower
triangular -- and also easily invertible by negating the subdiagonal
entries., just as in Gaussian elimination without pivoting. Writing

we have the desired

- Permute the rows of A using P.
- Apply Gassian elimination
*without pivoting*to PA.

Note that scaling the rows of A to choose the pivot point does not change this since the choice of the pivot point is a separate issue.