Elementary Row Operation/Examples

Examples of Elementary Row Operations

Example: $r_2 \to \lambda r_2$

Consider the elementary row operation $e$ defined as:

$e := r_2 \to \lambda r_2$

acting on a matrix space $\map \MM {3, n}$ for some $n \in \Z_{>0}$.

The elementary row matrix corresponding to $e$ is:

$\begin {pmatrix} 1 & 0 & 0 \\ 0 & \lambda & 0 \\ 0 & 0 & 1 \end {pmatrix}$

Example: $r_3 \to r_3 + 2 r_2$

Consider the elementary row operation $e$ defined as:

$e := r_3 \to r_3 + 2 r_2$

acting on a matrix space $\map \MM {3, n}$ for some $n \in \Z_{>0}$.

The elementary row matrix corresponding to $e$ is:

$\begin {pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 2 & 1 \end {pmatrix}$

Example: $r_1 \leftrightarrow r_2$

Consider the elementary row operation $e$ defined as:

$e := r_1 \leftrightarrow r_2$

acting on a matrix space $\map \MM {3, n}$ for some $n \in \Z_{>0}$.

The elementary row matrix corresponding to $e$ is:

$\begin {pmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end {pmatrix}$

Arbitrary Operation on Identity Matrix

The matrix:

$\mathbf A = \begin {pmatrix} 1 & 0 \\ 1 & 1 \end {pmatrix}$

can be obtained from the identity matrix $\mathbf I_2$ by the elementary row operation $e$ defined as:

$e := r_2 \to r_1 + r_2$

Then multiplying the matrix:

$\mathbf X = \begin {pmatrix} a & b \\ c & d \end {pmatrix}$

on the left by $\mathbf A$ we get:

$\begin {pmatrix} a & b \\ a + c & b + d \end {pmatrix}$

which can be obtained by applying that same elementary row operation $e$ on $\mathbf X$.

Operations on Arbitrary Matrix

Let $\mathbf A$ be the matrix:

$\mathbf A = \begin {pmatrix} 1 & 2 & 3 & 4 \\ 2 & -1 & 1 & 0 \\ -2 & 3 & 1 & 1 \end {pmatrix}$

Example: $r_2 \to \lambda r_2$

Let the elementary row operation $e$ be applied to $\mathbf A$, where $e$ is defined as:

$e := r_2 \to \lambda r_2$

Then $\mathbf A$ is transformed into:

$\mathbf A = \begin {pmatrix} 1 & 2 & 3 & 4 \\ 2 \lambda & -\lambda & \lambda & 0 \\ -2 & 3 & 1 & 1 \end {pmatrix}$

Example: $r_3 \to r_3 + 2 r_2$

Let the elementary row operation $e$ be applied to $\mathbf A$, where $e$ is defined as:

$e := r_3 \to r_3 + 2 r_2$

Then $\mathbf A$ is transformed into:

$\mathbf A = \begin {pmatrix} 1 & 2 & 3 & 4 \\ 2 & -1 & 1 & 0 \\ 2 & 1 & 3 & 1 \end {pmatrix}$

Example: $r_1 \leftrightarrow r_2$

Let the elementary row operation $e$ be applied to $\mathbf A$, where $e$ is defined as:

$e := r_1 \leftrightarrow r_2$

Then $\mathbf A$ is transformed into:

$\mathbf A = \begin {pmatrix} 2 & -1 & 1 & 0 \\ 1 & 2 & 3 & 4 \\ -2 & 3 & 1 & 1 \end {pmatrix}$