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
Best Answer
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 $PI=P$.