Formally Proving the Invariance of Solutions in Gaussian Elimination

[adartsesirhc]

I've been doing Gaussian Elimination in a Linear Algebra class, but I have a question:

How do I formally prove that elementary row operations do not change the set of solutions to a system of linear equations?

Thanks.

 

[quasar987]

Instead of using matrix representation for the system of equation, keep it in equation form.

Then elementary row operations are the same as either permuting two equations (obviously doesn't change the solutions!), multiplying one equations by a constant (doesn't change the solutions as you can probably easily see), and multipling an equation by a constant and adding the resulting equation to another. If you think about it for a second, you'll see why this doesn't change the solutions either.