There is a fundamental connection between systems of linear equations and certain matrix equations. Indeed, given a system of linear equations
\[ \begin{array}{cccccccccc} \colorbox{lightgreen}{$a_{11}$} {\color{red} x_1} & + & \colorbox{lightgreen}{$a_{12}$} {\color{red}x_2} & + & \cdots & \cdots & + & \colorbox{lightgreen}{$a_{1n}$} {\color{red} x_n} & = & c_1 \\ \colorbox{lightgreen}{$a_{21}$} {\color{red} x_1} & + & \colorbox{lightgreen}{$a_{22}$} {\color{red}x_2} & + & \cdots & \cdots & + & \colorbox{lightgreen}{$a_{2n}$} {\color{red} x_n} & = & c_2 \\ \vdots & & \vdots & & \ddots & & \vdots & && \vdots \\ \vdots & & \vdots & & & \ddots & \vdots & && \vdots \\ \colorbox{lightgreen}{$a_{m1}$} {\color{red} x_1} & + & \colorbox{lightgreen}{$a_{m2}$} {\color{red}x_2} & + & \cdots & \cdots & + & \colorbox{lightgreen}{$a_{mn}$} {\color{red} x_n} & = & c_m \\ \end{array} \]form its associated
Coefficient Matrix | Variables Vector | Augmentation Vector |
| \[ A \DefEq \left[ \begin{array}{ccc} a_{11} & \cdots & a_{1n} \\ \vdots & \ddots & \vdots \\ a_{m1} & \cdots & a_{mn} \end{array} \right] \] | \[ {\color{red} X \DefEq \left[ \begin{array}{c} x_1 \\ \vdots \\ x_n \end{array} \right] } \] | \[ C \DefEq \left[ \begin{array}{c} c_1 \\ \vdots \\ c_m \end{array} \right] \] |
Then observe that the left hand side of the given system of linear equations is just the matrix product
\[ \left[ \begin{array}{ccc} a_{11} & \cdots & a_{1n} \\ \vdots & \ddots & \vdots \\ a_{m1} & \cdots & a_{mn} \end{array} \right] \cdot {\color{red} \left[ \begin{array}{c} x_1 \\ \vdots \\ x_n \end{array} \right] }\ =\ \left[ \begin{array}{ccccc} \colorbox{lightgreen}{$a_{11}$} {\color{red} x_1} & + & \cdots & + & \colorbox{lightgreen}{$a_{1n}$} {\color{red} x_n} \\ \vdots & & & & \vdots \\ \colorbox{lightgreen}{$a_{m1}$} {\color{red} x_1} & + & \cdots & + & \colorbox{lightgreen}{$a_{mn}$} {\color{red} x_n} \end{array} \right] \]Accordingly, the given system of linear equations is equivalent to the matrix equation
\[\Mtrx{A}{\color{red} \Mtrx{X}} = \Mtrx{C}\]Example The system of linear equatons
\[ \begin{array}{rcccrcr} \colorbox{lightgreen}{$2$} {\color{red} x} & + & \colorbox{lightgreen}{$3$} {\color{red} y} & - & \colorbox{lightgreen}{$1$} {\color{red} z} & = & 6 \\ \colorbox{lightgreen}{$4$} {\color{red} x} & - & \colorbox{lightgreen}{$2$} {\color{red} z} & + & \colorbox{lightgreen}{$6$} {\color{red} z} & = & 2 \end{array} \]is equivalent to the matrix equation
\[ \colorbox{lightgreen}{$\left[\begin{array}{rrr} 2 & 3 & -1 \\ 4 & -2 & 6 \end{array}\right]$}\cdot {\color{red} \begin{bmatrix} x \\ y \\ z \end{bmatrix} }\ =\ \begin{bmatrix} 6 \\ 2 \end{bmatrix} \]