We have that an elementary row operation $e$ is used to transform $\begin {pmatrix} \mathbf A & \mathbf b \end {pmatrix}$ to $\begin {pmatrix} \mathbf A' & \mathbf b' \end {pmatrix}$. Now, whatever $e$ is, $\begin {pmatrix} \mathbf A' & \mathbf b' \end {pmatrix}$ is the augmented matrix of a system of simultaneous linear equations $S'$. We investigate each type of elementary row operation in turn. In the below, let: :$r_k$ denote row $k$ of $\mathbf A$ :$r'_k$ denote row $k$ of $\mathbf A'$ for arbitrary $k$ such that $1 \le k \le m$. By definition of elementary row operation, only the row or rows directly operated on by $e$ is or are different between $\begin {pmatrix} \mathbf A & \mathbf b \end {pmatrix}$ and $\begin {pmatrix} \mathbf A' & \mathbf b' \end {pmatrix}$. Hence it is understood that in the following, only those equations corresponding to those rows directly affected will be under consideration. === $\text {ERO} 1$: Scalar Product of Row === Let $e \begin {pmatrix} \mathbf A & \mathbf b \end {pmatrix}$ be the elementary row operation: :$e := r_k \to \lambda r_k$ where $\lambda \ne 0$. Then the equation in $S$: :$(1 \text a): \displaystyle \sum_{i \mathop = 1}^n \alpha_{k i} x_i = \beta_k$ is replaced in $S'$ by: {{begin-eqn}} {{eqn | l = \sum_{i \mathop = 1}^n \lambda \alpha_{k i} x_i       | r = \lambda \beta_k       | c =  }} {{eqn | n = 2a       | ll= \leadsto       | l = \lambda \sum_{i \mathop = 1}^n \alpha_{k i} x_i       | r = \lambda \beta_k       | c =  }} {{end-eqn}} It is seen that $\tuple {x_1, x_2, \ldots x_n}$ is a solution to $(1 \text a)$ {{iff}} $\tuple {x_1, x_2, \ldots x_n}$ is a solution to $(2 \text a)$. {{qed|lemma}} === $\text {ERO} 2$: Add Scalar Product of Row to Another === Let $e \begin {pmatrix} \mathbf A & \mathbf b \end {pmatrix}$ be the elementary row operation: :$e := r_k \to r_k + \lambda r_l$ Then the equation in $S$: :$(1 \text b): \displaystyle \sum_{i \mathop = 1}^n \alpha_{k i} x_i = \beta_k$ is replaced in $S'$ by: {{begin-eqn}} {{eqn | l = \sum_{i \mathop = 1}^n \paren {\alpha_{k i} + \lambda \alpha_{l i} } x_i       | r = \beta_k + \lambda \beta_l       | c =  }} {{eqn | n = 2b       | ll= \leadsto       | l = \sum_{i \mathop = 1}^n \alpha_{k i} x_i + \lambda \sum_{i \mathop = 1}^n \alpha_{l i} x_i       | r = \lambda \beta_k + \lambda \beta_l       | c =  }} {{end-eqn}} It is seen that $\tuple {x_1, x_2, \ldots x_n}$ is a solution to $(1 \text b)$ {{iff}} $\tuple {x_1, x_2, \ldots x_n}$ is a solution to $(2 \text b)$. {{qed|lemma}} === $\text {ERO} 3$: Exchange Rows === Let $e \begin {pmatrix} \mathbf A & \mathbf b \end {pmatrix}$ be the elementary row operation: :$e := r_k \leftrightarrow r_l$ Then the equations in $S$: {{begin-eqn}} {{eqn | n = 1c       | l = \sum_{i \mathop = 1}^n \alpha_{k i} x_i       | r = \beta_k       | c =  }} {{eqn | n = 2c       | l = \sum_{i \mathop = 1}^n \alpha_{l i} x_i       | r = \beta_l       | c =  }} {{end-eqn}} exist unchanged in $S'$, but are in different positions. It follows trivially that: :$\tuple {x_1, x_2, \ldots x_n}$ is a solution to $(1 \text c)$ in $S$ {{iff}}: :$\tuple {x_1, x_2, \ldots x_n}$ is a solution to $(1 \text c)$ in $S'$ and: :$\tuple {x_1, x_2, \ldots x_n}$ is a solution to $(2 \text c)$ in $S$ {{iff}}: :$\tuple {x_1, x_2, \ldots x_n}$ is a solution to $(2 \text c)$ in $S'$. {{qed|lemma}} Thus in all cases, for each elementary row operation which transforms $\begin {pmatrix} \mathbf A & \mathbf b \end {pmatrix}$ to $\begin {pmatrix} \mathbf A' & \mathbf b' \end {pmatrix}$, $S$ is equivalent to $S'$. Finally we note that from Existence of Inverse Elementary Row Operation, there exists an elementary row operation $e'$ which transforms $\begin {pmatrix} \mathbf A' & \mathbf b' \end {pmatrix}$ to $\begin {pmatrix} \mathbf A & \mathbf b \end {pmatrix}$. Hence, mutatis mutandis, the above argument can be used to demonstrate that $S'$ is equivalent to $S$. Hence the result. {{qed}}