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For the sake of commodity, from now on each linear system will have an associated matrix. For now, we may think a matrix as a box of numbers; it has of course a deep meaning that will be studied later

$$ \left\{ \begin{array}{rrrrrrrcl} a_{11}x_1 & + & a_{12}x_2 & + & \cdots & + & a_{1n}x_n & = & b_1\\ a_{21}x_1 & + & a_{22}x_2 & + & \cdots & + & a_{2n}x_n & = & b_2\\ & & & & & & & \vdots & \\ a_{m1}x_1 & + & a_{m2}x_2 & + & \cdots & + & a_{mn}x_n & = & b_m\\ \end{array}\right. \qquad\longleftrightarrow\qquad \left(\begin{array}{cccc|c} a_{11} & a_{12} & \cdots & a_{1n} & b_1 \\ a_{21} & a_{22} & \cdots & a_{2n} & b_2 \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} & b_m \\ \end{array}\right) $$

and it is denoted $(A|b)$. For instance,

$$ \left\{ \begin{array}{rrrrr} x_1 & -2x_2 & + x_3 & = & 7\\ 2x_1 & -5x_2 & +2x_3 & = & 6\\ 3x_1 & 2x_2 & - x_3 & = & 1\\ \end{array}\right. \qquad\longleftrightarrow\qquad \left( \begin{array}{ccc|c} 1 & -2 & 1 & 7 \\ 2 & -5 & 2 & 6 \\ 3 & 2 & -1 & 1 \\ \end{array}\right)=(A|b) $$ $$A=\left( \begin{array}{ccc} 1 & -2 & 1 \\ 2 & -5 & 2 \\ 3 & 2 & -1 \\ \end{array}\right)\qquad\qquad\qquad b=\left( \begin{array}{c} 7 \\ 6 \\ 1 \\ \end{array}\right)$$