> [!NOTE] Lemma > Let $(A \mid \underline{b})$ be a [[Linear System of Equations|linear system]] of $m$ equations and let $E \in$ Mat $_{m m}$ be an [[Elementary Matrices|elementary matrix]]. Then $\underline{x}$ is a [[Solution to Linear System of Equations|solution]] of $(A \mid \underline{b})$ if and only if $\underline{x}$ is a solution of $(E A \mid E \underline{b})$. **Proof**: If $A \underline{x}=\underline{b}$ then, by [[Real Matrix Product|premultiplying]] by $E$, we have $E A \underline{x}=E \underline{b}$. By [[Elementary Matrices are Invertible]], $E^{-1}$ exists. Conversely, suppose $E A \underline{x}=E \underline{b}$, then premultiplying by $E^{-1}$ gives have $A \underline{x}=\underline{b}$.