Coefficient Matrix

Definition: Coefficient Matrix

The coefficient 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 $m\times n$-matrix $A\in F^{m\times n}$ whose entries are the coefficients before the unknowns:

$$A=\left[\begin{array}{ccc}{a}_{11}& \cdots & a_{1n}\\ ⋮& \ddots & ⋮\\ a_{m1}& \cdots & a_{mn}\end{array}\right]$$