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:
\(\ds \map e {\mathbf A}\) | \(=\) | \(\ds \mathbf E \mathbf A\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \begin {pmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end {pmatrix} \begin {pmatrix} 1 & 2 & 3 & 4 \\ 2 & -1 & 1 & 0 \\ -2 & 3 & 1 & 1 \end {pmatrix}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \begin {pmatrix} 2 & -1 & 1 & 0 \\ 1 & 2 & 3 & 4 \\ -2 & 3 & 1 & 1 \end {pmatrix}\) |
$\blacksquare$
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: Exercise $1.1$