Invertible matrix/Staircase form/Identity matrix/Fact/Proof

Proof

This rests on the manipulations of the elimination procedure, and on the fact that elementary row manipulations are achieved, due to fact, by multiplications with elementary matrices from the left. In doing this, it can not happen that a zero-column or a zero-row arises, because the elementary matrices are invertible, and, in each step, invertibility is preserved. If we have an invertible upper triangular matrix, then the diagonal entries are not ${}0$ by exercise, and, by multiplication with a scalar, we can normalize them to ${}1$. With this, we can further achieve, in every column, that all entries above the diagonal entry are ${}0$.

To fact