Elementary Matrix is Invertible

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

Then $\mathbf E_n$ is invertible.

Proof
Let $\sim$ denote row equivalence.

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

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

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

Define:

We have:


 * $\mathbf E_1 \mathbf E_1' = \mathbf E_1' \mathbf E_1 = \mathbf I$.

Next, let $\mathbf E_2$ be an elementary matrix created by the operation $r_i \to r_i + a r_j$.

Then $r_i \to r_i - a r_j$ will undo this operation.

Define $\mathbf E_2'$ as:

Then:


 * $\mathbf E_2 \mathbf E_2' = \mathbf E_2' \mathbf E_2 = \mathbf I$

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

This operation undoes itself, and:


 * $\mathbf E_3 \mathbf E_3 = \mathbf I$

Hence the result, from Proof by Cases.