3. For every natural integer $n$, we denote by $C _ { n }$ the column matrix $\binom { u _ { n + 1 } } { u _ { n } }$. We denote by $A$ the square matrix of order 2 such that, for every natural integer $n$, $C _ { n + 1 } = A C _ { n }$. Determine $A$ and prove that, for every natural integer $n , C _ { n } = A ^ { n } C _ { 0 }$. 4. Let $P = \left( \begin{array} { l l } 2 & 3 \\ 1 & 1 \end{array} \right) , D = \left( \begin{array} { l l } 2 & 0 \\ 0 & 3 \end{array} \right)$ and $Q = \left( \begin{array} { c c } - 1 & 3 \\ 1 & - 2 \end{array} \right)$. Calculate $Q P$. It is admitted that $A = P D Q$. Prove by induction that, for every non-zero natural integer $n , A ^ { n } = P D ^ { n } Q$.
Let $\mathscr { P } _ { 1 }$ be the plane with equation $x + y + z = 0$ and $\mathscr { P } _ { 2 }$ be the plane with equation $x + 4 y + 2 = 0$.
3. For every natural integer $n$, we denote by $C _ { n }$ the column matrix $\binom { u _ { n + 1 } } { u _ { n } }$.
We denote by $A$ the square matrix of order 2 such that, for every natural integer $n$, $C _ { n + 1 } = A C _ { n }$.\\
Determine $A$ and prove that, for every natural integer $n , C _ { n } = A ^ { n } C _ { 0 }$.\\
4. Let $P = \left( \begin{array} { l l } 2 & 3 \\ 1 & 1 \end{array} \right) , D = \left( \begin{array} { l l } 2 & 0 \\ 0 & 3 \end{array} \right)$ and $Q = \left( \begin{array} { c c } - 1 & 3 \\ 1 & - 2 \end{array} \right)$.
Calculate $Q P$.\\
It is admitted that $A = P D Q$.\\
Prove by induction that, for every non-zero natural integer $n , A ^ { n } = P D ^ { n } Q$.\\