---
tag: 0P7A2MJD24
kind: definition
name: Coefficient matrix
msc: [15A06]
defines: 0HQFZAP3ND
uses: [0JTRMBQ1SA, 0N1MY1T0W9]
sources:
  - url: https://en.wikipedia.org/wiki/Coefficient_matrix
    locator: lead section
---

## Definition

Let $F$ be a field.
Take a [[system of linear equations|0GT4WWGWH2]] over $F$ with $m$ equations in $n$ unknowns, where equation $i$ is $a_{i1} x_1 + \cdots + a_{in} x_n = b_i$.
The *coefficient matrix* of the system is the $m \times n$ [[matrix|0VD9ZSWWJG]] $A = (a_{ij})$.
Its entry in row $i$ and column $j$ is the coefficient of $x_j$ in equation $i$.

## Remarks

The system $2x + y = 5$, $x - 3y = 0$ has the coefficient matrix $\begin{pmatrix} 2 & 1 \\ 1 & -3 \end{pmatrix}$.
With $x = (x_1, \dots, x_n)^T$ and $b = (b_1, \dots, b_m)^T$, the system is $Ax = b$.
If $A$ is square and [[invertible|0EB9N0S33C]], the system has exactly one solution, $x = A^{-1} b$.

## Explanations

### L1 human

The coefficient matrix keeps only the numbers in front of the unknowns, in a rectangular table.
Each row is one equation, and each column is one unknown.
This table lets us work with the whole system as one object.