---
tag: 0YCDZ7N8VD
kind: definition
name: Augmented matrix
msc: [15A06]
defines: 0T6MWGZSXA
uses: [0P7A2MJD24]
sources:
  - url: https://en.wikipedia.org/wiki/Augmented_matrix
    locator: lead section
---

## Definition

Let $F$ be a field.
Take a [[system of linear equations|0GT4WWGWH2]] $Ax = b$ over $F$, with [[coefficient matrix|0HQFZAP3ND]] $A$ of size $m \times n$ and constant vector $b \in F^m$.
The *augmented matrix* of the system is the $m \times (n+1)$ matrix $[A \mid b]$.
Its first $n$ columns are the columns of $A$, and its last column is $b$.

## Remarks

The system $2x + y = 5$, $x - 3y = 0$ has the augmented matrix $\left(\begin{array}{cc|c} 2 & 1 & 5 \\ 1 & -3 & 0 \end{array}\right)$.
[[Elementary row operations|04XWH28452]] on the augmented matrix do not change the solution set of the system.
More generally, $[A \mid B]$ denotes the matrix $A$ with the columns of a matrix $B$ added on the right.

## Explanations

### L1 human

The augmented matrix is the full system written as a table of numbers.
The vertical bar reminds us where the equals signs were.
We do row reduction on this table instead of rewriting the equations each time.