Update matrix entries in beamer

One possibility is to use array environments with columns of type p{<length>}, instead of bmatrix:

\documentclass{beamer}
\usepackage{array}

\begin{document}

\begin{frame}{Row-echelon form}
  Reduce the following matrix to row-echelon form.
  \[
\left[
\begin{array}{*{4}{>{\raggedleft\arraybackslash$}p{1.3em}<{$}}}
    2 & 1 & -1 & 3 \\
    1 & -1 & 2 & 1 \\
    -4 & 6 & -7 & 1 \\
    2 & 0 & 1 & 3
  \end{array}
\right]
  \onslide<2->
  \to
\left[
  \begin{array}{*{4}{>{\raggedleft\arraybackslash$}p{1,3em}<{$}}}
    1 & -1 & 2 & 1 \\
    \alt<2>{2}{0} & \alt<2>{1}{3} & \alt<2>{-1}{-5} & \alt<2>{3}{1} \\
    \alt<2-3>{-4}{0} & \alt<2-3>{6}{2} & \alt<2-3>{-7}{1} & \alt<2-3>{1}{5} \\
    \alt<2-4>{2}{0} & \alt<2-4>{0}{2} & \alt<2-4>{1}{-3} & \alt<2-4>{3}{1}
  \end{array}
\right]
\]
  \action<5>{}
\end{frame}

\end{document}

You can avoid that the entries move around by measuring the widths of the two alternatives and using the larger of the two widths. For this I use \mathmakeboxlargestof from this nice answer of Philippe Goutet.

\documentclass{beamer}
\usepackage{calc}
\usepackage{mathtools}
\newlength{\letterwidth}
\newcommand{\mathmakeboxlargestof}[3]{%
    \setlength{\letterwidth}{\maxof{\widthof{$#1$}}{\widthof{$#2$}}}%
    \mathmakebox[\letterwidth]{\vphantom{#1}\vphantom{#2}#3}%
}
\makeatletter
\newcommand\replace{}
\def\replace<#1>#2#3{%
    \alt<#1>{\mathmakeboxlargestof{#2}{#3}{{#2}}}%
            {\mathmakeboxlargestof{#2}{#3}{{#3}}}%
    }
\makeatother

\begin{document}

\begin{frame}{Row-echelon form}
  Reduce the following matrix to row-echelon form.
  \[\begin{bmatrix}
    2 & 1 & -1 & 3 \\
    1 & -1 & 2 & 1 \\
    -4 & 6 & -7 & 1 \\
    2 & 0 & 1 & 3
  \end{bmatrix}
  \onslide<2->
  \to
  \begin{bmatrix}
    1 & -1 & 2 & 1 \\
    \replace<2>{2}{0} & \replace<2>{1}{3} & \replace<2>{-1}{-5} & \replace<2>{3}{1} \\
    \replace<2-3>{-4}{0} & \replace<2-3>{6}{2} & \replace<2-3>{-7}{1} & \replace<2-3>{1}{5} \\
    \replace<2-4>{2}{0} & \replace<2-4>{0}{2} & \replace<2-4>{1}{-3} & \replace<2-4>{3}{1}
  \end{bmatrix}\]
  \action<5>{}
\end{frame}

\end{document}

Note that I use additional braces {} around the entries #2 and #3; otherwise the spacing of the minus signs wouldn't be correct.