Definition:Elementary Operation/Row

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Definition

Let $\mathbf A = \sqbrk a_{m n}$ be an $m \times n$ matrix over a field $K$.

The elementary row operations on $\mathbf A$ are operations which act upon the rows of $\mathbf A$ as follows.


For some $i, j \in \closedint 1 m: i \ne j$:

\((\text {ERO} 1)\)   $:$   \(\ds r_i \to \lambda r_i \)    For some $\lambda \in K_{\ne 0}$, multiply row $i$ by $\lambda$             
\((\text {ERO} 2)\)   $:$   \(\ds r_i \to r_i + \lambda r_j \)    For some $\lambda \in K$, add $\lambda$ times row $j$ to row $i$             
\((\text {ERO} 3)\)   $:$   \(\ds r_i \leftrightarrow r_j \)    Exchange rows $i$ and $j$             


Also defined as

The order of presentation of the elementary matrix operations, either row or column, may vary according to the source.

Some sources use the Greek letter $\rho$ to enumerate the rows, and $\kappa$ to enumerate the columns, and jocularly remind us that the name rho of the letter $\rho$ is pronounced row.


Examples

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:

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


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:

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


Also see

  • Results about elementary row operations can be found here.


Sources