Can You Use Non-Pivot Rows 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.

 

[tacman]

Hello,

Thank you for sharing your experience with scaling matrices. It is true that the Gauss algorithm is commonly used to scale a matrix, but there are other methods that can also be used.

One alternative method is to use row operations, such as multiplying a row by a constant or adding or subtracting rows, to transform the matrix into an upper triangular form. This can be done by choosing any pivot element, not necessarily the first one, and performing row operations until all the elements below the pivot are zero.

Another approach is the LU decomposition method, which involves decomposing the original matrix into a lower triangular matrix and an upper triangular matrix. This can be useful for solving systems of equations and can also be used for scaling a matrix.

Ultimately, the choice of method depends on the specific problem and the desired outcome. While the Gauss algorithm may be the most commonly used, it is not the only option and it is important to be open to different approaches. As long as the final solution is correct, the method used to obtain it should not matter.