Invertible matrix/Find inverse matrix/Table/Method

Let ${}M$ denote a square matrix. How can we decide whether the matrix is invertible, and how can we find the inverse matrix ${}M^{-1}$?

For this we write down a table, on the left-hand side we write down the matrix ${}M$, and on the right-hand side we write down the identity matrix (of the right size). Now we apply on both sides step by step the same elementary row manipulations. The goal is to produce in the left-hand column, starting with the matrix, in the end the identity matrix. This is possible if and only if the matrix is invertible. We claim that we produce, by this method, in the right column the matrix ${}M^{-1}$ in the end. This rests on the following invariance principle. Every elementary row manipulation can be realized as a matrix multiplication with some elementary matrix ${}E$ from the left. If in the table we have somewhere the pair

$(M_{1},M_{2}),$

after the next step (in the next line) we have

$(EM_{1},EM_{2}).$

If we multiply the inverse of the second matrix (which we do not know yet; however, we do know its existence, in case the matrix is invertible) with the first matrix, then we get

${}(EM_{1})^{-1}EM_{2}=M_{1}^{-1}E^{-1}EM_{2}=M_{1}^{-1}M_{2}\,.$

This means that this expression is not changed in each single step. In the beginning, this expression equals ${}M^{-1}E_{n}$, hence in the end, the pair ${}(E_{n},N)$ must fulfil

${}N=E_{n}^{-1}N=M^{-1}E_{n}=M^{-1}\,.$