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

== Theorem ==
Let $S$ be a system of simultaneous linear equations:
:$\ds \forall i \in \set {1, 2, \ldots, m}: \sum_{j \mathop = 1}^n \alpha_{i j} x_j = \beta_i$
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.
=== Corollary 1 ===
=== Corollary 2 ===
=== Corollary 3 ===
=== Corollary 4 ===
=== Corollary 5 ===
== 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 $S$:
:$(1 \text a): \ds \sum_{i \mathop = 1}^n \alpha_{k i} x_i = \beta_k$
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 $S$:
:$(1 \text b): \ds \sum_{i \mathop = 1}^n \alpha_{k i} x_i = \beta_k$
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 $S'$
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 $S'$.
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.
== Sources ==
* : Chapter $1$: Linear Equations and Matrices: $1.3$ Applications to Linear Equations: Theorem $1.7$
Category:Elementary Row Operations
Category:Simultaneous Linear Equations
Category:Elementary Row Operation on Augmented Matrix leads to Equivalent System of Simultaneous Linear Equations