# Coefficient matrix

In linear algebra, the coefficient matrix refers to a matrix consisting of the coefficients of the variables in a set of linear equations. The matrix is used in solving systems of linear equations.

## Coefficient matrix

In general, a system with m linear equations and n unknowns can be written as

${\displaystyle a_{11}x_{1}+a_{12}x_{2}+\cdots +a_{1n}x_{n}=b_{1}\,}$
${\displaystyle a_{21}x_{1}+a_{22}x_{2}+\cdots +a_{2n}x_{n}=b_{2}\,}$
${\displaystyle \vdots \,}$
${\displaystyle a_{m1}x_{1}+a_{m2}x_{2}+\cdots +a_{mn}x_{n}=b_{m}\,}$

where ${\displaystyle x_{1},\ x_{2},...,x_{n}}$ are the unknowns and the numbers ${\displaystyle a_{11},\ a_{12},...,\ a_{mn}}$ are the coefficients of the system. The coefficient matrix is the mxn matrix with the coefficient ${\displaystyle a_{ij}}$ as the (i,j)-th entry:[1]

${\displaystyle {\begin{bmatrix}a_{11}&a_{12}&\cdots &a_{1n}\\a_{21}&a_{22}&\cdots &a_{2n}\\\vdots &\vdots &\ddots &\vdots \\a_{m1}&a_{m2}&\cdots &a_{mn}\end{bmatrix}}}$

## References

1. ^ Liebler, Robert A. (December 2002). Basic Matrix Algebra with Algorithms and Applications. CRC Press. pp. 7–8. Retrieved 13 May 2016.