@JosephWright Am I missing something?
\documentclass{beamer}
\usepackage[ngerman]{babel}
\begin{document}
\begin{frame}
    \begin{proof}
    \end{proof}
    \begin{corollary}
    \end{corollary}
    \begin{example}
    \end{example}
\end{frame}
\end{document}

\documentclass{beamer}
\usepackage[ngerman]{babel}
\providetranslation{Corollary}{Korollar}
\providetranslation{Example}{Beispiel}
\begin{document}
\begin{frame}
    \begin{proof}
    \end{proof}
    \begin{corollary}
    \end{corollary}
    \begin{example}
    \end{example}
\end{frame}
\end{document}

\documentclass{article}
\let\matrix\relax
\makeatletter
\newenvironment{matrix}
 {\let\\\cr\null\,\vcenter\bgroup\normalbaselines\m@th\ialign\bgroup
  \hfil$##$\hfil&&\quad\hfil$##$\hfil\crcr\mathstrut\crcr\noalign{\kern-\baselineskip}}
 {\crcr\mathstrut\crcr\noalign{\kern-\baselineskip}\egroup\egroup\,}
\makeatother
\begin{document}
$\left(\begin{matrix}a & b\\ c & d\end{matrix}\right)$
\end{document}

\documentclass{article}

\begin{document}

\[
\Bigl( \frac{1+x}{1-x} \Bigr)
\qquad
\biggl( \frac{1+x}{1-x} \biggr)
\]

\[
\Bigl( \frac{ 1+A^{x^2-1} }{ 1-A^{x^2-1} } \Bigr)
\qquad
\biggl( \frac{ 1+A^{x^2-1} }{ 1-A^{x^2-1} } \biggr)
\]

\end{document}

\documentclass{article}

\begin{document}

\[
\Bigl( \frac{1+x}{1-x} \Bigr)
+
\biggl( \frac{ 1+A^{x^2-1} }{ 1-A^{x^2-1} } \biggr)
\]

\[
\biggl( \frac{1+x}{1-x} \biggr)
+
\biggl( \frac{ 1+A^{x^2-1} }{ 1-A^{x^2-1} } \biggr)
\]

\end{document}