Invertibility of Elementary Matrix

Theorem

Let $E$ : Any elementary matrix

Then

• $E$ is invertible
• $E^{-1}$ is an also an elementary matrix

Proof

If $E$ is an elementary matrix, then $E$ results by performing some elementary row operation operation on $I$.

Let $E_0$ be the matrix that results when the inverse of this operation is performed on $I$.

Applying theorem Row Operations by Matrix Multiplications, and using the fact that inverse row operations cancel the effect of each other, it follows that

$$E_{0}E=E{E}_0=I$$

Thus, by definition of inverse matrix, $E_0$ is the inverse of $E_0$