Augmented matrix

Definition

Let $F$ be a field. Take a system of linear equations $Ax = b$ over $F$, with coefficient matrix $A$ 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 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.