Elementary Row Operation/Examples/Operations on Arbitrary Matrix/Swap r1 and r2

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Example of Elementary Row Operation

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:

$e := r_1 \leftrightarrow r_2$

Then $\mathbf A$ is transformed into:

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


Proof

From Elementary Row Operation: $r_1 \leftrightarrow r_2$, the elementary row matrix $\mathbf E$ corresponding to $e$ is:

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

Hence by Elementary Row Operations as Matrix Multiplications:

\(\ds \map e {\mathbf A}\) \(=\) \(\ds \mathbf E \mathbf A\)
\(\ds \) \(=\) \(\ds \begin {pmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end {pmatrix} \begin {pmatrix} 1 & 2 & 3 & 4 \\ 2 & -1 & 1 & 0 \\ -2 & 3 & 1 & 1 \end {pmatrix}\)
\(\ds \) \(=\) \(\ds \begin {pmatrix} 2 & -1 & 1 & 0 \\ 1 & 2 & 3 & 4 \\ -2 & 3 & 1 & 1 \end {pmatrix}\)

$\blacksquare$


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