Elementary Row Operations and Preserving Solutions.

[Bacle]

Hi Again:

Just curious: I know that, given a system of linear equations,

ERO's (scaling both sides of an equation, exchanging/swapping rows

and adding a multiple of a row to another row) preserve solutions,

i.e., if x is a solution to Ax=b, then swapping rows will preserve

x as a solution, and no other solution will pop up, and same for other

two. More specifically, if given a system S, we use its associated

augmented matrix A, with x a solution to Ax=b and we swap rows

to get a matrix A', then A'x=b ; same for the other two Elementary Row

Ops.

This above is not so hard to show, but:

* question* how do we know it is precisely these three operations--

and no others--that preserve solutions?

Thanks.

 

[fresh_42]

We have $Ax=b$. Now given a regular matrix $T$ we can write $A'=TAx=Tb=b'$and get a new system with the same solutions. This means any regular $T$ is allowed. By a step by step transformation this will be the matrices
$$
\begin{bmatrix}1&c\\0&1\end{bmatrix}\, , \,\begin{bmatrix}0&1\\1&0\end{bmatrix}\, , \,\begin{bmatrix}\frac{1}{c}&0\\0&1\end{bmatrix}
$$
expanded with the identity matrix elsewhere to match the dimensions.