What's the best way make an "augmented" coefficient matrix?

Question

When studying systems of linear equations, it's nice to remind people that the last column of the coefficient matrix holds the constants. This is often done in books by putting a vertical line between the last column and the next to last column. What is a good way to do this in LaTex?

Answer 1

One way to do this is implemented in the (free, in both senses!) online linear algebra textbook Linear Algebra by Jim Hefferon. It's written in LaTeX and is open-source so one can download the book and its attendant style files. One of them, called linalgjh.sty is about typesetting common linear algebra stuff such as augmented matrices and row reductions and the like. The code for the augmented matrices is:

\newenvironment{amatrix}[1]{%
  \left(\begin{array}{@{}*{#1}{c}|c@{}}
}{%
  \end{array}\right)
}

and is used as:

\begin{amatrix}{2}
   1 & 2 & 3 \\  a & b & c
 \end{amatrix}

(note that the argument is one less than the total number of columns). I guess that the @{}s at the start and end are to get the spacing right with the parentheses (mentioned by TH in a comment to fabikw's similar answer).

That style file has several other useful linear algebra macros that may be useful.

Answer 2

If you build your own environment using array, you're on the safe side. I would extend an internal macro of amsmath using an optional argument.

Advantages:

• It extends several matrix environments at the same time (matrix, pmatrix, bmatrix, Bmatrix, vmatrix, Vmatrix).

• The names and meanings of those environments remain (not apmatrix etc.)

• Spacing etc. is the same like in amsmath.

• You could do more than just insert a vertical line (use color and alignment, for instance right aligned columns because of minus signs).

• If you omit the optional argument, it acts exactly like the amsmath environment.

Caution:

• Since you redefine an internal macro, it might not work if the original package changes its code. But amsmath.sty has not been changed for more than 10 years. If there's a change in the matrices later, you could adjust your own macro.

Code:

Here's the redefinition, just put it in your preamble after loading amsmath:

\makeatletter
\renewcommand*\env@matrix[1][*\c@MaxMatrixCols c]{%
  \hskip -\arraycolsep
  \let\@ifnextchar\new@ifnextchar
  \array{#1}}
\makeatother

I had to use \makeatletter ... \makeatother because of the @ in macro names. The optional argument is the standard amsmath argument to \array. The original definition in amsmath is:

\def\env@matrix{\hskip -\arraycolsep
  \let\@ifnextchar\new@ifnextchar
  \array{*\c@MaxMatrixCols c}}

So, there's not much that's been changed.

Examples:

Simple augmented matrix:

\begin{pmatrix}[cc|c]
  1 & 2 & 3\\
  4 & 5 & 9
\end{pmatrix}

More complex use, with different alignment, spacing and color:

\begin{bmatrix}[*2cr@{\quad}|@{\quad}>{\color{red}}r]
  a & b & 1  &  4 \\
  c & d & -2 & -3
\end{bmatrix}

Output:

Just remove the optional argument in brackets and you will get the standard pmatrix or bmatrix.

I've used this code and examples in a blog post in 2008: An extension to amsmath matrix environments.

If you don't wish to redefine that internal macro, you could give it a different name and build your amatrix environment on it exactly the same way that amsmath does with pmatrix.

Answer 3

If you are using an array to input the matrix, you just have to specify something like

\left(\begin{array}{cc|c}  
 2 & 0 & 1\\  
 0 & 1 & 1  
\end{array}\right)

Answer 4

If you would rather not override styles, you can create an augmented matrix using standard matrix commands like so:

\begin{pmatrix}
    1 & 0 & 0 & 0 &\bigm| & 0 \\
    0 & 1 & 0 & 0 &\bigm| & 5 \\
    0 & 0 & 1 & 0 &\bigm| & -4 \\ 
    0 & 0 & 0 & 1 &\bigm| & -2
\end{pmatrix}

generating the following.

Answer 5

If for some reason (e.g. you try to make augmented matrix in Pages) other answers do not work, you can make an augmented matrix with two matrices inside delimiters of your choice:

\left[
  \begin{matrix}
    1 & 2 & 3 \\
    1 & 2 & 3 \\
    1 & 2 & 3 \\
    1 & 2 & 3 \\
  \end{matrix}
  \left|
    \,
    \begin{matrix}
      4  \\
      4  \\
      4  \\
      4  \\
    \end{matrix}
  \right.
\right]

This code produces a matrix like this:

You can modify space between delimiters by adding \, or other spacers.

Answer 6

The easybmat package provides a way to add all sorts of lines between cells in a matrix, including dashed lines which I seem to recall being popular in my undergraduate linear algebra text.

Edit: Here's an example. I'm actually not thrilled about the spacing.

\[\left(\begin{BMAT}(@){cc.c}{cc}
a_{11} & a_{12} & b_1\\
a_{21} & a_{22} & b_2
\end{BMAT}\right)\]

Answer 7

The package nicematrix provides tools to draw mathematical matrices. There is no environment for augmented matrices in this package but it's easy to create one (compatible with the other features of nicematrix: dotted lines, blocks, exterior rows and columns, etc.).

\documentclass{article}
\usepackage{nicematrix}
\usepackage{tikz}

\makeatletter
\ExplSyntaxOn
\NewDocumentEnvironment { pNiceMatrixAug } { }
  { \begin { pNiceMatrix } }
  { 
    \CodeAfter \tikz \draw (1-|\arabic{jCol}) -- (last-|\arabic{jCol}) ; 
    \end { pNiceMatrix } 
  }

\ExplSyntaxOff
\makeatother

\begin{document}

$\begin{pNiceMatrixAug}
a_{11} & a_{12} & \Cdots & a_{1n} & 0 \\
a_{21} & a_{22} & \Cdots & a_{2n} & 0 \\
\Vdots &       &        & \Vdots & \Vdots \\
a_{n1} & a_{n2} & \Cdots & a_{nn} & 0 \\
\end{pNiceMatrixAug}$

\end{document}

However, you need several compilations.

---

EDIT: In order to have the ability to use \CodeBefore (and \Body) and \CodeAfter as in all the environment of nicematrix, and a color of the vertical rule which respects the key rules\color (and \arrayrulecolor), we must write an environment {pNiceMatrixAux} a bit more complicated:

\documentclass{article}
\usepackage{nicematrix}
\usepackage{tikz}

\makeatletter
\ExplSyntaxOn
\NewDocumentEnvironment { pNiceMatrixAug } { }
  { \pNiceMatrix } 
  { 
    \tl_gput_right:Nn \g_nicematrix_code_after_tl
      { 
        \group_begin:
        \CT@arc@ % for the color of the rule
        \tikz \draw (1-|\arabic{jCol}) -- (last-|\arabic{jCol}) ; 
        \group_end:
      }
    \endpNiceMatrix
  }
\ExplSyntaxOff
\makeatother

\begin{document}

$\begin{pNiceMatrixAug}[rules/color=blue,left-margin]
\CodeBefore 
  \cellcolor{red!15}{1-1,2-2,4-4}
\Body  
a_{11} & a_{12}  & \Cdots & a_{1n} & 0 \\
a_{21} & a_{22}  & \Cdots & a_{2n} & 0 \\
\Vdots & \Vdots &        & \Vdots & \Vdots \\
a_{n1} & a_{n2}  & \Cdots & a_{nn} & 0 \\
\CodeAfter
  \line{2-2}{4-4}
\end{pNiceMatrixAug}$

\end{document}

Answer 8

I would do something like this:

{
\centering
$$
\left(
\begin{array}{ccc|c}
1 & 1 & 1 & 1 \\ 
8 & 4 & 2 & 1 \\ 
64 & 16 & 4 & 1 \\ 
    \end{array}
\right)
$$
}

Answer 9

A simple and clean solution for an augmented matrix can be obtained with the tabularray package.

\documentclass{article}

\usepackage{tabularray}

\NewDocumentEnvironment{Abmatrix}{+b}{
\left[
\begin{tblr}{rowsep = {1pt}, colsep = {5pt}, column{1} = {leftsep = 0pt}, column{Z} = {rightsep = 0pt}, cells = {c}, vline{Y} = {dashed, 0.75pt}}
#1
\end{tblr}
\right]
}{}

\begin{document}

An augmented bmatrix:

\[
\begin{Abmatrix}
2 & -1 & 2 & 9 \\
1 & -2 & 4 & 12 \\
3 & 1 & -1 & 2
\end{Abmatrix}
\]

And another one:

\[
\begin{Abmatrix}
3 & 5 & 7 & 8 & 3 \\
6 & -2 & -3 & 12 & 5
\end{Abmatrix}
\]
\end{document}

Answer 10

\left[\begin{array}{cccc|c}
1 & 0 & 1 & -1 & 0 \\
-1 & 1 & -2 & -1 & 0 \\
1 & -2 & 3 & 2 & 0
\end{array}\right]