> [!NOTE] Proposition (Elementary matrices are invertible) > The [[Inverse of Real Square Matrix|inverse]] of [[Elementary Matrices|elementary matrices]] exists. *Proof*. Observe that $\begin{align}(S_{ij})^{-1}&=S_{ji}=S_{ij}\\ (M_{i}(\lambda))^{-1} &= M_{i}\left( \frac{1}{\lambda} \right) \\ (A_{ij}(\lambda))^{-1} &= A_{ij}(-\lambda) \end{align}$