Augmented Matrix

Definition: Augmented Matrix

The augmented matrix of a system of linear equations

$$|\begin{array}{rl}{a}_{11}{x}_1+\cdots +a_{1n}{x}_n& =b_1\\ a_{21}{x}_1+\cdots +a_{2n}{x}_n& =b_2\\ ⋮⋮& \\ a_{m1}{x}_1+\cdots +a_{mn}{x}_n& =b_n\end{array}\text{ where }{a}_{ij},b_i\in F$$

is the coefficient matrix $A$ of the system with an additional column containing the values $b_1,\cdots ,b_n$ on the right-hand side of the equations:

$$\left[\begin{array}{cccc}{a}_{11}& \cdots & a_{1n}& b_1\\ ⋮& \ddots & ⋮& ⋮\\ a_{m1}& \cdots & a_{mn}& b_n\end{array}\right]$$

NOTATION

$$(A\mid \vec{b})\left[\begin{array}{cccc}{a}_{11}& \cdots & a_{1n}& b_1\\ ⋮& \ddots & ⋮& ⋮\\ a_{m1}& \cdots & a_{mn}& b_n\end{array}\right]$$