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The following operations (called elementary operations) lead to equivalent systems of equations:

  1. Swapping rows $i$ and $j$

    $$ \left\{ \begin{array}{rrrrrrrcl} a_{11}x_1 & + & a_{12}x_2 & + & \cdots & + & a_{1n}x_n & = & b_1\\ & & & & & & & \vdots & \\ a_{i1}x_1 & + & a_{i2}x_2 & + & \cdots & + & a_{in}x_n & = & b_i\\ & & & & & & & \vdots & \\ a_{j1}x_1 & + & a_{j2}x_2 & + & \cdots & + & a_{jn}x_n & = & b_j\\ & & & & & & & \vdots & \\ a_{m1}x_1 & + & a_{m2}x_2 & + & \cdots & + & a_{mn}x_n & = & b_m\\ \end{array}\right. \qquad\leadsto\qquad \left\{ \begin{array}{rrrrrrrcl} a_{11}x_1 & + & a_{12}x_2 & + & \cdots & + & a_{1n}x_n & = & b_1\\ & & & & & & & \vdots & \\ a_{j1}x_1 & + & a_{j2}x_2 & + & \cdots & + & a_{jn}x_n & = & b_j\\ & & & & & & & \vdots & \\ a_{i1}x_1 & + & a_{i2}x_2 & + & \cdots & + & a_{in}x_n & = & b_i\\ & & & & & & & \vdots & \\ a_{m1}x_1 & + & a_{m2}x_2 & + & \cdots & + & a_{mn}x_n & = & b_m\\ \end{array}\right. $$

    $$ \left(\begin{array}{cccc|c} a_{11} & a_{12} & \cdots & a_{1n} & b_1 \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ a_{i1} & a_{i2} & \cdots & a_{in} & b_i \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ a_{j1} & a_{j2} & \cdots & a_{jn} & b_j \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} & b_m \\ \end{array}\right) \qquad\leadsto\qquad \left(\begin{array}{cccc|c} a_{11} & a_{12} & \cdots & a_{1n} & b_1 \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ a_{j1} & a_{j2} & \cdots & a_{jn} & b_j \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ a_{i1} & a_{i2} & \cdots & a_{in} & b_i \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} & b_m \\ \end{array}\right) $$

  2. Multiplying row $i$ by a nonzero scalar $k$

    $$ \left\{ \begin{array}{rrrrrrrcl} a_{11}x_1 & + & a_{12}x_2 & + & \cdots & + & a_{1n}x_n & = & b_1\\ & & & & & & & \vdots & \\ a_{i1}x_1 & + & a_{i2}x_2 & + & \cdots & + & a_{in}x_n & = & b_i\\ & & & & & & & \vdots & \\ a_{m1}x_1 & + & a_{m2}x_2 & + & \cdots & + & a_{mn}x_n & = & b_m\\ \end{array}\right. \qquad\leadsto\qquad \left\{ \begin{array}{rrrrrrrcl} a_{11}x_1 & + & a_{12}x_2 & + & \cdots & + & a_{1n}x_n & = & b_1\\ & & & & & & & \vdots & \\ ka_{i1}x_1 & + & ka_{i2}x_2 & + & \cdots & + & ka_{in}x_n & = & kb_i\\ & & & & & & & \vdots & \\ a_{m1}x_1 & + & a_{m2}x_2 & + & \cdots & + & a_{mn}x_n & = & b_m\\ \end{array}\right. $$

    $$ \left(\begin{array}{cccc|c} a_{11} & a_{12} & \cdots & a_{1n} & b_1 \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ a_{i1} & a_{i2} & \cdots & a_{in} & b_i \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} & b_m \\ \end{array}\right) \qquad\leadsto\qquad \left(\begin{array}{cccc|c} a_{11} & a_{12} & \cdots & a_{1n} & b_1 \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ ka_{i1} & ka_{i2} & \cdots & ka_{in} & kb_i \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} & b_m \\ \end{array}\right) $$

  3. Adding $k$ times a row $i$ to a row $j$

    $$ \left\{ \begin{array}{rrrrrrrcl} a_{11}x_1 & + & a_{12}x_2 & + & \cdots & + & a_{1n}x_n & = & b_1\\ & & & & & & & \vdots & \\ a_{i1}x_1 & + & a_{i2}x_2 & + & \cdots & + & a_{in}x_n & = & b_i\\ & & & & & & & \vdots & \\ a_{j1}x_1 & + & a_{j2}x_2 & + & \cdots & + & a_{jn}x_n & = & b_j\\ & & & & & & & \vdots & \\ a_{m1}x_1 & + & a_{m2}x_2 & + & \cdots & + & a_{mn}x_n & = & b_m\\ \end{array}\right. \qquad\leadsto\qquad \left\{ \begin{array}{rrrrrrrcl} a_{11}x_1 & + & a_{12}x_2 & + & \cdots & + & a_{1n}x_n & = & b_1\\ & & & & & & & \vdots & \\ a_{i1}x_1 & + & a_{i2}x_2 & + & \cdots & + & a_{in}x_n & = & b_i\\ & & & & & & & \vdots & \\ (a_{j1}+ka_{i1})x_1 & + & (a_{j2}+ka_{i2})x_2 & + & \cdots & + & (a_{jn}+ka_{in})x_n & = & (b_j+kb_i)\\ & & & & & & & \vdots & \\ a_{m1}x_1 & + & a_{m2}x_2 & + & \cdots & + & a_{mn}x_n & = & b_m\\ \end{array}\right. $$

    $$ \left(\begin{array}{cccc|c} a_{11} & a_{12} & \cdots & a_{1n} & b_1 \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ a_{i1} & a_{i2} & \cdots & a_{in} & b_i \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ a_{j1} & a_{j2} & \cdots & a_{jn} & b_j \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} & b_m \\ \end{array}\right) \qquad\leadsto\qquad \left(\begin{array}{cccc|c} a_{11} & a_{12} & \cdots & a_{1n} & b_1 \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ a_{i1} & a_{i2} & \cdots & a_{in} & b_i \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ a_{j1}+ka_{i1} & a_{j2}+ka_{i2} & \cdots & a_{jn}+ka_{in} & b_j+kb_i \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} & b_m \\ \end{array}\right) $$