Requirements/Conditions for Gauss Jordan elimination/Gaussian elimination

Question

I'm currently studying methods to solve systems of linear equations. With the Gauss Jordan elimination, I was wondering if there were any requirements or conditions required for the method to work with a system.

Answer 1

No, one of the beauty of Gauss-Jordan elimination is that you can always apply it, as long as you work over the real $\mathbb{R}$ or complex numbers $\mathbb{C}.$

Answer 2

The general requirement is that we work over a field, i.e. all non-zero elements have an inverse.
Once you have that, the Gauss Jordan elimination will work for any matrix; assuming, of course, that you can actually do computations in said field.

There are variants of the algorithm for different rings, but I am not aware of a general version that works for all rings; it's more like "in this special ring, we have property X, which helps us to do the algorithm as follows...".