Gauss-Jordan elimination is a technique that can be used to calculate the inverse of matrices (if they are invertible). It can also be used to solve simultaneous linear equations.
However, after a few google searches, I have failed to find a proof that this algorithm works for all $n \times n$, invertible matrices. How would you prove that the technique of using Gauss-Jordan elimination to calculate matrices will work for all invertible matrices of finite dimensions (we allow swapping of two rows)?
Induction on $n$ is a possible idea: the base case is very clear, but how would you prove the inductive step?
We are not trying to show that an answer generated using Gauss-Jordan will be correct. We are trying to show that Gauss-Jordan can apply to all invertible matrices.
Note: I realize that there is a similar question here, but this question is distinct in that it asks for a proof for invertible matrices.