Bonjour Charline,
Exercice 71
[tex]A=\begin{pmatrix}0&\dfrac{1}{2}\\\\-1&\dfrac{1}{2}\end{pmatrix}\\\\\\Det(A)=0\times\dfrac{1}{2}+1\times\dfrac{1}{2}=\dfrac{1}{2}\\\\A^{-1}=\dfrac{1}{\dfrac{1}{2}}\begin{pmatrix}\dfrac{1}{2}&-\dfrac{1}{2}\\\\1&0\end{pmatrix}\\\\\\A^{-1}=2\begin{pmatrix}\dfrac{1}{2}&-\dfrac{1}{2}\\\\1&0\end{pmatrix}\\\\\\\Longrightarrow\boxed{A^{-1}=\begin{pmatrix}1&-1\\\\2&0\end{pmatrix}}[/tex]
Preuve
[tex]AA^{-1}=\begin{pmatrix}0&\dfrac{1}{2}\\\\-1&\dfrac{1}{2}\end{pmatrix}\begin{pmatrix}1&-1\\\\2&0\end{pmatrix}=\begin{pmatrix}0+1&0+0\\\\-1+1&1+0\end{pmatrix}=\begin{pmatrix}1&0\\\\0&1\end{pmatrix}\\\ \ et\\A^{-1}A=\begin{pmatrix}1&-1\\\\2&0\end{pmatrix}\begin{pmatrix}0&\dfrac{1}{2}\\\\-1&\dfrac{1}{2}\end{pmatrix}=\begin{pmatrix}0+1&\dfrac{1}{2}-\dfrac{1}{2}\\\\0+0&1+0\end{pmatrix}=\begin{pmatrix}1&0\\\\0&1\end{pmatrix}\\\\\\\Longrightarrow\boxed{AA^{-1}=A^{-1}A=I_2}[/tex]
Exercice 72
[tex]A=\begin{pmatrix}1&-1\\2&0\end{pmatrix}\\\\\\\Longrightarrow\boxed{A^{-1}=\begin{pmatrix}0&\dfrac{1}{2}\\\\-1&\dfrac{1}{2}\end{pmatrix}}[/tex]
Exercice 73
[tex]A=\begin{pmatrix}1&1\\\\-1&1\end{pmatrix}\\\\\\\Longrightarrow\boxed{A^{-1}=\begin{pmatrix}\dfrac{1}{2}&-\dfrac{1}{2}\\\\\dfrac{1}{2}&\dfrac{1}{2}\end{pmatrix}}[/tex]
Exercice 74
[tex]A=\begin{pmatrix}2&4\\\\\dfrac{1}{2}&3\end{pmatrix}\\\\\\\Longrightarrow\boxed{A^{-1}=\begin{pmatrix}\dfrac{3}{4}&-1\\\\-\dfrac{1}{8}&\dfrac{1}{2}\end{pmatrix}}[/tex]
