Elementary Row Operation as Sequence of Other Two

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Theorem

Let $\mathbf A$ be an $m \times n$ matrix.

Let $i, j \in \left[{1 . . m}\right]: i \ne j$

Let $r_i = \left({\mathbf A_{i,1}, \mathbf A_{i,2}, \cdots, \mathbf A_{i,n}}\right)$, and let $r_j$ be defined likewise.

Then the elementary row operation $r_i \leftrightarrow r_j$ can be written as a finite sequence of exactly four of the other two types of elementary row operations.


Proof

We start with $\left({r_i, \; r_j}\right)$

  • $r_j \to r_j + r_i: \quad \left({r_i, \; r_i + r_j}\right)$
  • $r_i \to r_i + \left({-r_j}\right): \quad \left({-r_j, \; r_i + r_j}\right)$
  • $r_j \to r_j + r_i: \quad \left({-r_j, \; r_i}\right)$

And finally,

  • $r_i \to -r_i: \quad \left({r_j, \; r_i}\right)$


$\blacksquare$