Bonjour,
[tex]1)\\
A=\left[\ \begin{array}{cc}a&b\\0&a\end{array}\right]\\\\A^2=A*A=\left[\ \begin{array}{cc}a&b\\0&a\end{array}\right]*\left[\ \begin{array}{cc}a&b\\0&a\end{array}\right]=\left[\ \begin{array}{cc}a^2&2ab\\0&a^2\end{array}\right]\\\\A^3=A*A^2=\left[\ \begin{array}{cc}a&b\\0&a\end{array}\right]*\left[\ \begin{array}{cc}a^2&2ab\\0&a^2\end{array}\right]=\left[\ \begin{array}{cc}a^3&3a^2b\\0&a^3\end{array}\right]\\\\[/tex]
[tex]A^4=A*A^3=\left[\ \begin{array}{cc}a&b\\0&a\end{array}\right]*\left[\ \begin{array}{cc}a^3&3a^2b\\0&a^3\end{array}\right]=\left[\ \begin{array}{cc}a^4&4a^3b\\0&a^4\end{array}\right]\\\\[/tex]
2)
[tex]\boxed{A^n=\left[\ \begin{array}{cc}a^n&na^{n-1}b\\0&a^n\end{array}\right]\\}[/tex]
Démonstration par réccurrence.
[tex]A^5=A*A^4=\left[\ \begin{array}{cc}a&b\\0&a\end{array}\right]*\left[\ \begin{array}{cc}a^4&4a^3b\\0&a^4\end{array}\right]=\left[\ \begin{array}{cc}a^5&5a^4b\\0&a^5\end{array}\right]\\\\[/tex]
