Augmented Matrix¶

Definition¶

The augmented matrix of a linear system $\boldsymbol{A}\boldsymbol{x}=\boldsymbol{b}$ is the matrix formed by adjoining the right-hand side vector $\boldsymbol{b}$ to the coefficient matrix $\boldsymbol{A}$ , often written $(\boldsymbol{A}\mid \boldsymbol{b})$ $b)$ .

Interpretation¶

An augmented matrix is a linear system with the names of the unknowns, the plus signs and the equals signs removed: each row is one equation, each column to the left of the bar belongs to one unknown, and the last column holds the right-hand sides. It contains all numerical information needed to solve the system.

A system of three linear equations next to its augmented matrix, with the coefficient block A separated by a vertical line from the right-hand side column b.

The figure turns the system $2x_1+x_2-x_3=8$ $8$ , $-3x_1-x_2+2x_3=-11$ $- 11$ , $-2x_1+x_2+2x_3=-3$ into a $3\times4$ matrix. The shaded $3\times3$ block is the coefficient matrix $\boldsymbol{A}$ , copied equation by equation, and the column after the bar is $\boldsymbol{b}=(8,-11,-3)$ $(8, - 11, - 3)$ .

Writing the system this way makes row operations convenient: an operation on equations becomes an operation on rows, and because $\boldsymbol{b}$ is part of every row, the right-hand sides change together with the coefficients. After reduction to row echelon form the solution set can be read off; a row $(0\;\cdots\;0\mid c)$ with $c\neq0$ stands for the equation $0=c$ $c$ , so the system has no solution. This test is later restated in terms of rank.

A missing unknown must be entered as $0$ in its column, so that every column stays with one unknown. The name should also not be confused with the augmented form of a linear program in OR Modelling, which adds slack variables to the model rather than a column to a matrix.

Key Results¶

Elementary row operations on the augmented matrix correspond to elementary operations on the equations.
Gauss-Jordan elimination is performed on augmented matrices.

Examples¶

For $2x+y=3$ $3$ , $x-y=1$ $1$ , the augmented matrix is $\begin{pmatrix}2&1&3\\1&-1&1\end{pmatrix}$ .
The vertical bar in $(\boldsymbol{A}\mid \boldsymbol{b})$ is visual; the augmented matrix is still a matrix.

Prerequisites¶

Sources¶

Math II: Linear Algebra · Week 3

Aliases: augmented coefficient matrix, augmented system matrix

Last updated: 20 October 2026 03:02