System of Linear Equations in Matrix Form

A system of $m$ linear equations in $n$ unknowns,

can be packaged as a single matrix equation. Let $A=[a_{ij}]$ be the $m\times n$ coefficient matrix (row $i$ = the coefficients of equation $i$, in the fixed unknown order $x_1,\ldots,x_n$), $X=\begin{pmatrix}x_1\\\vdots\\x_n\end{pmatrix}$ the $n\times1$ column of unknowns, and $B=\begin{pmatrix}b_1\\\vdots\\b_m\end{pmatrix}$ the $m\times1$ column of constants. Then the entire system is equivalent to the single matrix equation

The matrix $[A\,|\,B]$ (append the constants column to $A$) is the augmented matrix of the system. …