Transformation of Unit Matrix into Inverse

Theorem
Let $\mathbf{A}$ be a square matrix of order $n$.

Let $\mathbf{I}$ be the identity matrix of order $n$.

Suppose there exists a sequence of elementary row operations that reduces $\mathbf{A}$ to to $\mathbf{I}$.

Then and only then is $\mathbf{A}$ invertible.

Futhermore, the same sequence, when performed on $\mathbf{I}$, results in the inverse of $\mathbf{A}$.

Proof
For ease of presentation, let $\breve{\mathbf{X}}$ be the inverse of $\mathbf{X}$.

We have that $\mathbf{A}$ can be transformed into $\mathbf{I}$ by a sequence of elementary row operations.

By repeated application of Elementary Row Operations by Matrix Multiplication, we can write this assertion as:

Because each elementary matrix is invertible, we can multiply on the left both sides of this equation by:

By repeated application of Elementary Row Operations by Matrix Multiplication, each $\mathbf{E_n}$ on the right hand side corresponds to an elementary row operation.

Hence the result.