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
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The question is a bit strange, since seeing the process is also showing why it works ! In a first descending pass you create iteratively more and more zero coefficients on the left, resulting on a triangle of non-zero coefs. Then in a ascending pass you iteratively identify the rightest unknown and go up (or you could also just create iteratively more and more zero coefficient on the right, letting only the diagonal, which gives a system trivial to solve).
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 $PI=P$.