Elementary Row Operation/Examples/Swap r1 and r2

Examples of Elementary Row Operations

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:

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

Proof

By definition, the elementary row matrix corresponding to $e$ is found by applying $e$ to the unit matrix.

By definition of unit matrix:

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

Let $\mathbf E$ denote the elementary row matrix corresponding to $e$.

$\mathbf E$ is constructed by exchanging row $1$ with row $2$.

$\blacksquare$

Sources

• 1998: Richard Kaye and Robert Wilson: Linear Algebra ... (previous) ... (next): Part $\text I$: Matrices and vector spaces: $1$ Matrices: $1.5$ Row and column operations: $3$