Transformation of Unit Matrix into Inverse

Theorem
Let $\mathbf A$ be a square matrix of order $n$ of the matrix space $\mathbf M_n \left({\R}\right)$.

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 $\mathbf A$ is 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.