Bonjour Charline661
Exercice 56
[tex]R=\begin{pmatrix}-3&3&-1\\-6&7&-4\\-5&6&-4\end{pmatrix}[/tex]
[tex]1)a)\ R^2=R\times R\\\\R^2=\begin{pmatrix}-3&3&-1\\-6&7&-4\\-5&6&-4\end{pmatrix}\times\begin{pmatrix}-3&3&-1\\-6&7&-4\\-5&6&-4\end{pmatrix}\\\\\\R^2=\begin{pmatrix}a_{11}&a_{12}&a_{13}\\a_{21}&a_{22}&a_{23}\\a_{31}&a_{32}&a_{33}\end{pmatrix}\\\\\\a_{11}=-3\times(-3)+3\times(-6)-1\times(-5)=-4\\a_{12}=-3\times3+3\times7-1\times6=6\\a_{13}=-3\times(-1)+3\times(-4)-1\times(-4)=-5\\a_{21}=-6\times(-3)+7\times(-6)-4\times(-5)=-4\\a_{22}=-6\times3+7\times7-4\times6=7[/tex]
[tex]a_{23}=-6\times(-1)+7\times(-4)-4\times(-4)=-6\\a_{31}=-5\times(-3)+6\times(-6)-4\times(-5)=-1\\a_{32}=-5\times3+6\times7-4\times6=3\\a_{33}=-5\times(-1)+6\times(-4)-4\times(-4)=-3\\\\\\\Longrightarrow\boxed{R^2=\begin{pmatrix}-4&6&-5\\-4&7&-6\\-1&3&-3\end{pmatrix}}[/tex]
b) Une démarche analogue nous montrerait que :
[tex]R^3=R^2\rimes R\\\\R^3=\begin{pmatrix}-4&6&-5\\-4&7&-6\\-1&3&-3\end{pmatrix}\times\begin{pmatrix}-3&3&-1\\-6&7&-4\\-5&6&-4\end{pmatrix}\\\\\\R^3=\begin{pmatrix}1&0&0\\0&1&0\\0&0&1\end{pmatrix}\\\\\\\Longrightarrow\boxed{R^3=I_3}[/tex]
[tex]2)\ R^4=R^3\times R=I_3\times R=R\\\\\Longrightarrow\boxed{R^4=\begin{pmatrix}-3&3&-1\\-6&7&-4\\-5&6&-4\end{pmatrix}}\\\\\\R^5=R^3\times R^2=I_3\times R^2=R^2\\\\\Longrightarrow\boxed{R^5=\begin{pmatrix}-4&6&-5\\-4&7&-6\\-1&3&-3\end{pmatrix}}\\\\\\R^6=R^3\times R^3=I_3\times I_3=I_3\\\\\Longrightarrow\boxed{R^6=\begin{pmatrix}1&0&0\\0&1&0\\0&0&1\end{pmatrix}}[/tex]
