Elementary Row Operation on Augmented Matrix leads to Equivalent System of Simultaneous Linear Equations

Theorem
Let $S$ be a system of simultaneous linear equations:


 * $\displaystyle \forall i \in \set {1, 2, \ldots, m}: \sum_{j \mathop = 1}^n \alpha_{i j} x_j = \beta_i$

Let $\begin {pmatrix} \mathbf A & \mathbf b \end {pmatrix}$ denote the augmented matrix of $S$.

Let $\begin {pmatrix} \mathbf A' & \mathbf b' \end {pmatrix}$ be obtained from $\begin {pmatrix} \mathbf A & \mathbf b \end {pmatrix}$ by means of an elementary row operation.

Let $S'$ be the system of simultaneous linear equations of which $\begin {pmatrix} \mathbf A' & \mathbf b' \end {pmatrix}$ is the augmented matrix.

Then $S$ and $S'$ are equivalent.

Proof
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:

It is seen that $\tuple {x_1, x_2, \ldots x_n}$ is a solution to $(1 \text a)$ $\tuple {x_1, x_2, \ldots x_n}$ is a solution to $(2 \text a)$.

$\text {ERO} 2$: 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:

It is seen that $\tuple {x_1, x_2, \ldots x_n}$ is a solution to $(1 \text b)$ $\tuple {x_1, x_2, \ldots x_n}$ is a solution to $(2 \text b)$.

$\text {ERO} 3$: 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 $S$:

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 $S$


 * $\tuple {x_1, x_2, \ldots x_n}$ is a solution to $(1 \text c)$ in $S'$
 * $\tuple {x_1, x_2, \ldots x_n}$ 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 $S$


 * $\tuple {x_1, x_2, \ldots x_n}$ is a solution to $(2 \text c)$ in $S'$.
 * $\tuple {x_1, x_2, \ldots x_n}$ is a solution to $(2 \text c)$ in $S'$.

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.