Explanation of Gauss-Jordan elimination method. I know how to solve the system of linear equations, how to find inverse of matrix etc. by the Gauss-Jordan method.
But I want to understand why this method works (in cases of inverse matrix especially). Can you please explain in details why it works?
Thank you

Performing an elementary row operation on a  matrix $A$ amounts to left multiplying $A$ by a special type of (invertible) matrix. Thus performing a series of elementary row operations amounts to left multiplying by the product $P$ of these matrices, so that $PA=I$. Thus $P=A^{-1}$.
Performing the same operations on $I\;$ leads to $PI=P$.
