Bonjour Miss210B
Exercice 47 :
[tex]A\times B=\begin{pmatrix}\dfrac{2}{3} & 1\\\\-\dfrac{1}{2}
& \dfrac{2}{5}\end{pmatrix}\times\begin{pmatrix}
0 & \dfrac{2}{5} & 1\\\\
1 & -\dfrac{2}{5} & 0\end{pmatrix}\\\\\\A\times B=\begin{pmatrix}
\frac{2}{3}\times0+1\times1 & \frac{2}{3}\times\frac{2}{5}+1\times(-\frac{2}{5}) & \frac{2}{3}\times1+1\times0\\\\
-\frac{1}{2}\times0+\frac{2}{5}\times1 & -\frac{1}{2}\times\frac{2}{5}+\frac{2}{5}\times(-\frac{2}{5}) & -\frac{1}{2}\times1+\frac{2}{5}\times0\end{pmatrix}
[/tex]
[tex]A\times B=\begin{pmatrix} 0+1 & \dfrac{4}{15}-\dfrac{2}{5} & \dfrac{2}{3}+0\\\\ 0+\dfrac{2}{5} & -\dfrac{1}{5}-\dfrac{4}{25} & -\dfrac{1}{2}+0\end{pmatrix}\\\\\\\boxed{A\times B=\begin{pmatrix} 1 & -\dfrac{2}{15} & \dfrac{2}{3}\\\\ \dfrac{2}{5} & -\dfrac{9}{25} & -\dfrac{1}{2}\end{pmatrix}}[/tex]
Exercice 49
[tex]A\times B=\begin{pmatrix}0
& 1 & 0 & 0\\ 0
&0 & 1 & 0\\ 0
& 0 & 0 & 1\\ 4
& 0 & 3 & 0
\end{pmatrix}\times\begin{pmatrix}0
& -\dfrac{3}{4} & 0 & \dfrac{1}{4}\\ 1
&0 & 0 & 0\\ 0
& 1 & 0 & 0\\ 0
& 0 & 1 & 0
\end{pmatrix}[/tex]
[tex]\boxed{A\times B=\begin{pmatrix}
1 & 0 & 0 & 0\\
0 & 1 & 0 & 0\\
0 & 0 & 1 & 0\\
0 & 0 & 0 & 1
\end{pmatrix}}[/tex]
Exercice 55
[tex]1) a)\ \begin{pmatrix}-\dfrac{1}{2} & \dfrac{\sqrt{3}}{2}\\\\
-\dfrac{\sqrt{3}}{2} & -\dfrac{1}{2}
\end{pmatrix}\times\begin{pmatrix}-\dfrac{1}{2} & \dfrac{\sqrt{3}}{2}\\\\
-\dfrac{\sqrt{3}}{2} & -\dfrac{1}{2}
\end{pmatrix}=[/tex]
[tex]\begin{pmatrix}(-\dfrac{1}{2})\times(-\dfrac{1}{2})+\dfrac{\sqrt{3}}{2}\times(-\dfrac{\sqrt{3}}{2}) &&& -\dfrac{1}{2}\times\dfrac{\sqrt{3}}{2}+\dfrac{\sqrt{3}}{2}\times(-\dfrac{1}{2})\\\\
(-\dfrac{\sqrt{3}}{2})\times(-\dfrac{1}{2})-\dfrac{1}{2}\times(-\dfrac{\sqrt{3}}{2}) &&& (-\dfrac{\sqrt{3}}{2})\times\dfrac{\sqrt{3}}{2}+(-\dfrac{1}{2})\times(-\dfrac{1}{2})
\end{pmatrix}[/tex]
[tex]=\begin{pmatrix}\dfrac{1}{4}-\dfrac{3}{4} &&& -\dfrac{\sqrt{3}}{4}-\dfrac{\sqrt{3}}{4}\\\\
\dfrac{\sqrt{3}}{4}+\dfrac{\sqrt{3}}{4} &&& -\dfrac{3}{4}+\dfrac{1}{4}\end{pmatrix}\\\\\\=\begin{pmatrix}-\dfrac{1}{2} &&& -\dfrac{\sqrt{3}}{2}\\\\
\dfrac{\sqrt{3}}{2} &&& -\dfrac{1}{2}\end{pmatrix}[/tex]
[tex]b)\ R^3=R\times R^2=\begin{pmatrix}
1 &0 \\
0 & 1
\end{pmatrix}[/tex]
[tex]c)\ R^4=R\times R^3=R\times I_2=R\\\\R^5=R^2\times R^3=R^2\times I_2=R^2\\\\R^6=R^3\times R^3=I_2\times I_2=I_2[/tex]
