Elementary Row Operation/Examples/Operations on Arbitrary Matrix

Examples of Elementary Row Operations

Let $\mathbf A$ be the matrix:

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

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:

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

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:

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

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:

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