
\documentclass{article}
\usepackage[body={8cm,8cm}]{geometry}
\usepackage{lmodern}
\usepackage{mathtools}
\pagestyle{empty}
\newcommand{\transposee}[1]%
{{\vphantom{#1}}^{\mathit t}{#1}}
\begin{document}
\[
\transposee{
\begin{bmatrix}
   a & b \\ c & d \\
\end{bmatrix}
}
= 
\begin{bmatrix}
   a & c \\  b & d \\
\end{bmatrix}
\]
\end{document}
