I'm trying to prove that the Gauss-Jordan elimination works on any invertible matrix.
Suppose that $A\in\mathcal{M}_{n}(\mathbb R)$ is an invertible matrix. We construct the matrix $B\in\mathcal{M}_{n,2n}(\mathbb R)$ as follows: $$B=(\ A\ |\ I\ )$$ where $I\in\mathcal{M}_{n}(\mathbb R)$ is the identity matrix. We know that if we left multiply some matrices $E_1,E_2,E_3,...,E_p$ by $B$, we perform rows operations on $B$: $$E_p\cdot\ldots\cdot E_2\cdot E_1\cdot B=(\ I\ |\ C\ )$$ We get the matrix $(\ I\ |\ C\ )$. If we denote $E_p\cdot\ldots\cdot E_2\cdot E_1$ by $E$ (using the fact that the multiplication of matrices is associative), we write: $$E\cdot(\ A\ |\ I\ )=(\ I\ |\ C\ )$$ All the $E_i$ with $i=1, 2, \ldots, p$ and the matrix $E$, are $n\times n$ matrices. Since now, we've been multiplying the matrices considering them as block matrices. So we get that: $$(\ E A\ |\ EI\ ) = (\ I\ |\ C\ )$$ Now we deduce that $$\begin{cases} EA=I \\ EI=C \end{cases}$$ From the first equality we can conclude that $E=A^{-1}$ and, after replacing in the second equality, we get that $$C=A^{-1}$$ So, after operating on the matrix $B$, we get that $C$ must be equal to $A^{-1}$.
How do we know that the matrix $E$ we used in the proof, is invertible? The steps I've done to prove the Gauss-Jordan elimination, are correct?