Other ways to scale a matrix?

[Metame]

Hello, here's my problem : I was introduced the Gauss algorithm used to scale a matrix (to obtain a triangular matrix in order to solve a system of equations), but my teacher says that we can only use the pivot line to null the rest of the numbers of the column.

The fact is I have done quite a lot of exercises where I have to scale a matrix, but I don't necessaryly null with the pivot line, but any line (and often, by doing the right operations, I save a lot lot of time). She says I can't do that, even if I have the right solutions, and I don't see why. So why? I'm trying to demonstrate I can.

(Here's a very simple example to show what I'm talking about:

We have the matrix :
2 1 0 1 L1 first pivot line here
-1 2 0 1 L2
1 0 1 1 L3

Following carefully the Gauss algorithm :
2 1 0 1 L1
0 5 0 3 L2'=2*L2+L1 (forced to use L1, the pivot line)
0 -1 2 1 L3'=2*L3-L1 (forced to use L1 too)

Then L2' is the new pivot line, so I must do :
2 1 0 1 L1
0 5 0 3 L2
0 0 10 8 L3''=5*L3'+L2 (forced to use L2, pivot line)

and I have S={(1/5,3/5,4/5)})

 

[HallsofIvy]

If I remember correctly, the "pivot" is the largest number on the diagonal. You don't have to use that but, under some circumstances, using the largest number will increase the accuracy.