Elementary Matrix is Invertible

Theorem
Let $\mathbf{E_n}$ be an elementary matrix.

Then $\mathbf{E_n}$ is invertible.

Proof
Let $\mathbf{E_1}$ be an elementary matrix created by the operation $r_i \to ar_i$.

Then the operation $r_i \to \frac 1 a r_i$ will undo this operation, resulting in the identity matrix $\mathbf{I}$.

From Elementary Row Operations by Matrix Multiplication, for every elementary row operation there exists a corresponding elementary matrix.

Define:

where $\sim$ denotes row equivalence.

We have:


 * $\mathbf{E_1E_1'} = \mathbf{E_1'E_1} = \mathbf{I}$.

Next, let $\mathbf{E_2}$ be an elementary matrix created by the operation $r_i \to r_i + ar_j$.

Then $r_i \to r_i - ar_j$ will undo this operation.

Define $\mathbf{E_2'}$ as:

Then:


 * $\mathbf{E_2E_2'} = \mathbf{E_2'E_2} = \mathbf{I}$.

Lastly, let $\mathbf{E_3}$ be an elementary matrix created by the operation $r_i \leftrightarrow r_j$.

This operation undoes itself, and:


 * $\mathbf{E_3E_3} = \mathbf{I}$

Hence the result, from Proof by Cases.