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+%
+% inverse.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\begin{frame}[t]
+\frametitle{Inverse Matrix}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.24\textwidth}
+\begin{block}{Imaginäre Einheit}
+\vspace{-15pt}
+\begin{align*}
+J &= \begin{pmatrix} 0&-1\\1&0\end{pmatrix}
+\\
+0&=
+J^2 + I
+\\
+0&=
+J+J^{-1}
+\\
+J^{-1}&=-J
+\end{align*}
+\end{block}
+\end{column}
+\begin{column}{0.25\textwidth}
+\uncover<2->{%
+\begin{block}{Wurzel $\sqrt{2}$}
+\vspace{-15pt}
+\begin{align*}
+W&=\begin{pmatrix}0&2\\1&0\end{pmatrix}
+\\
+0 &= X^2-2
+\\
+0 &= W-2W^{-1}
+\\
+W^{-1}&=\frac12 W
+\end{align*}
+\end{block}}
+\end{column}
+\begin{column}{0.41\textwidth}
+\uncover<3->{%
+\begin{block}{Drehmatrix}
+\vspace{-15pt}
+\begin{align*}
+D&=\begin{pmatrix}
+\cos \frac{\pi}{1291} & -\sin\frac{\pi}{1291}\\
+\sin \frac{\pi}{1291} &  \cos\frac{\pi}{1291}
+\end{pmatrix}
+\\
+0 &= \only<-3>{D^{1291}+I\phantom{+\frac{\mathstrut}{\mathstrut}}}
+\only<4->{D^2-2D\cos\frac{\pi\mathstrut}{1291\mathstrut}+I}
+\\
+0 &= \only<-3>{D^{1290}+D^{-1}\phantom{+\frac{\mathstrut}{\mathstrut}}}
+\only<4->{D-2\cos\frac{\pi\mathstrut}{1291\mathstrut}+D^{-1}}
+\\
+D^{-1}
+&= \only<-3>{-D^{1290}\phantom{+\frac{\mathstrut}{\mathstrut}}}%
+\only<4->{-D+2I\cos\frac{\pi\mathstrut}{1291\mathstrut}}
+\end{align*}
+\end{block}}
+\end{column}
+\end{columns}
+\vspace{-25pt}
+\uncover<5->{
+\begin{block}{3D-Beispiel}
+$p(x) = -x^3-5x^2+5x+1$
+\[
+A=
+\begin{pmatrix*}[r]
+-5&-1&1\\
+-5&-2&3\\
+-1&-1&2
+\end{pmatrix*}
+\quad\Rightarrow\quad
+A^{-1}
+=
+A^2+5A-5I
+=
+\begin{pmatrix*}[r]
+-1& 1&-1\\
+ 7&-9&10\\
+ 3&-4& 5
+\end{pmatrix*}
+\]
+\end{block}}
+\vspace{-10pt}
+
+\end{frame}