For a system of linear equations, \begin{align*} a_{11}x_1+a_{12}x_2+a_{13}x_3+\dots+a_{1n}x_n&=b_1\\ a_{21}x_1+a_{22}x_2+a_{23}x_3+\dots+a_{2n}x_n&=b_2\\ a_{31}x_1+a_{32}x_2+a_{33}x_3+\dots+a_{3n}x_n&=b_3\\ \vdots&\\ a_{m1}x_1+a_{m2}x_2+a_{m3}x_3+\dots+a_{mn}x_n&=b_m \end{align*} the coefficient matrix is the $m\times n$ matrix \begin{equation*} A= \begin{bmatrix} a_{11}&a_{12}&a_{13}&\dots&a_{1n}\\ a_{21}&a_{22}&a_{23}&\dots&a_{2n}\\ a_{31}&a_{32}&a_{33}&\dots&a_{3n}\\ \vdots&\\ a_{m1}&a_{m2}&a_{m3}&\dots&a_{mn}\\ \end{bmatrix} \end{equation*}