The following paragraph is a description of the Gauss-Jordan-algorithm for determining the inverse of a matrix:
Description of the algorithm
"If A is an n × n square matrix, then one can use row reduction to compute its inverse matrix, if it exists. First, the n × n identity matrix is augmented to the right of A, forming an n × 2n block matrix [A | I]. Now through application of elementary row operations, find the reduced echelon form of this n × 2n matrix. The matrix A is invertible if and only if the left block can be reduced to the identity matrix I; in this case the right block of the final matrix is A−1. If the algorithm is unable to reduce the left block to I, then A is not invertible."
Question
It is easy to apply the algorithm. But why does it work, i.e. why does it yield a correct result? Can you explain why it guarantees correctness? It is easy to find sources that explain how to carry out the particular steps of the algorithm, but it seems that few bother to explain why the algorithm works.