Alignment within the equation environment

Question

This is the state of affairs:

The goal is to horizontally center the symbols over their expressions within an equation environment. So A is centered over the first matrix, x centered over the first vector and b centered over the final vector.

MWE:

\documentclass{article}
\usepackage{amsmath}

\begin{document}

  \begin{equation}
    \begin{split}
      %
      {\bf{A}} x & = b \\
      %
       \left[
       \begin{array}{cc}
          1 & 0 \\ 0 & 1 \\
       \end{array}
       \right]
       %
       \left[
         \begin{array}{c}
            x_1 \\ x_2
         \end{array}
       \right] &=
       %
       \left[
         \begin{array}{c}
            b_1 \\ b_2
         \end{array}
       \right]
       %
    \end{split}
  \end{equation}

\end{document}

These posts do not seem to offer readily a solution:

Complicated alignment within multiline equation

Aligning Multiline Matrix Equations

Aligning equations with text with alignat

Difference between align and alignat environments

TeXShop 3.58, distribution TeXLive - 2015

Answer 1

With array you can obtain the following:

\documentclass{article}
\usepackage{amsmath}
    
\begin{document}
\begin{equation}\arraycolsep=1pt
    \begin{array}{cccc}
\mathbf{A}      &  \mathbf{x}       & = &   \mathbf{b}  \\
    \begin{bmatrix}
    1 & 0 \\ 0 & 1 \\
    \end{bmatrix}   &   \begin{bmatrix}
                        x_1 \\ x_2
                        \end{bmatrix}   & = &   \begin{bmatrix}
                                                b_1 \\ b_2
                                                \end{bmatrix}
    \end{array}
\end{equation}
\end{document}

addendum: After seve years ...

Considered is @Mico comment below. Typographical more correct result you will get if in matrices be restored default \arraycolsep value (6pt). This can be done with adding

\AtBeginEnvironment{bmatrix}{\arraycolsep=6pt}

in document preamble. If you still have an older LaTeX installation, you also need to load \etoolbox package:

\documentclass{article}
\usepackage{mathtools}

%\usepackage{etoolbox} % activate if you have an older LaTeX instalation
\AtBeginEnvironment{bmatrix}{\arraycolsep=6pt}

% for show just the equation, don't use in the real document  ...
\usepackage[active,displaymath,tightpage]{preview}
\setlength\PreviewBorder{1em}

\begin{document}

\begin{equation}\arraycolsep=1pt
    \begin{array}{cccc}
\mathbf{A}      &   \mathbf{x}       & = &   \mathbf{b}  \\
\begin{bmatrix}
1 & 0 \\ 0 & 1 \\
\end{bmatrix}   &   \begin{bmatrix}
                    x_1 \\ x_2
                    \end{bmatrix}   & = &   \begin{bmatrix}
                                            b_1 \\ b_2
                                            \end{bmatrix}
    \end{array}
\end{equation}

\end{document}

Answer 2

The obvious choice is array, but with some tricks to avoid guessing the spaces.

\documentclass[twocolumn]{article}
\usepackage{amsmath}
\usepackage{array}

\newenvironment{lsarray}[1][c]% ls for 'linear system'
 {\begin{array}[#1]{@{} c @{} c @{} >{{}}c<{{}} @{} c @{}}}
 {\end{array}}

\begin{document}

Centered equation number:
\begin{equation}
\begin{lsarray}
  \mathbf{A} & \mathbf{x} & = & \mathbf{b} \\
  \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} &
  \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} &
  = &
  \begin{bmatrix} b_1 \\ b_2 \end{bmatrix}
\end{lsarray}
\end{equation}

Bottom equation number:
\begin{equation}
\begin{lsarray}[b]
  \mathbf{A} & \mathbf{x} & = & \mathbf{b} \\
  \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} &
  \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} &
  = &
  \begin{bmatrix} b_1 \\ b_2 \end{bmatrix}
\end{lsarray}
\end{equation}

\end{document}

The twocolumn option is used just to make a less wide picture, in order to better see the equation number placement in the two cases.