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\documentclass{scrartcl}
\usepackage{mathtools}% lädt amsmath

\begin{document}
\begin{equation}
\P^{\alpha,k} = 
\begin{cases}% genügt
\hfil\begin{bmatrix}
g \times n\\
n^k\\
\end{bmatrix}
& \alpha > k\\[4mm]
\hfil\begin{bmatrix}
\rho \times n^k\\
n^k\\
\end{bmatrix}
& \alpha = k\\[4mm]
\hfil\boldsymbol{0}   
& \alpha   < k\\                                 
\end{cases}
\end{equation}
\end{document}

\documentclass{scrartcl}
\usepackage{mathtools}

\begin{document}
\begin{equation}
\P^{\alpha,k} = 
\begin{dcases}
\begin{bmatrix}
g \times n\\
n^k\\
\end{bmatrix}
& \alpha > k\\
\begin{bmatrix}
\rho \times n^k\\
n^k\\
\end{bmatrix}
& \alpha = k\\
\boldsymbol{0}   
& \alpha   < k\\                                 
\end{dcases}
\end{equation}
\end{document}

\begin{equation}
\begin{aligned} 
\P^{\alpha,k} = \begin{cases}  \begin{bmatrix}
								        g \times n	\\
								        n^{k}	\\
						                    \end{bmatrix}
								   \alpha > k	\\
								     \begin{bmatrix}
							               \rho \times n^k	\\
							                n^{k}	\\
								     \end{bmatrix}
							            \alpha = k	\\
							           \boldsymbol{0}	
								     \alpha	< k \\										   
					    \end{cases}
\end{aligned} 
\end{equation}