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

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)

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)]

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@ifnextcharnew@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:

defenv@matrix{hskip -arraycolsep
  let@ifnextcharnew@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.

I would do something like this:

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

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.

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.