
Problema 2. Fie matricea $A=\left(\begin{array}{lll}2 & 1 & 1 \\ 1 & 2 & 1 \\ 1 & 1 & 2\end{array}\right) \in M_{3}(\mathbb{R})$. Să se calculeze $A^{2016}$.


Problema 2. Fie matricea $A=\left(\begin{array}{lll}2 & 1 & 1 \\ 1 & 2 & 1 \\ 1 & 1 & 2\end{array}\right) \in M_{3}(\mathbb{R})$. Să se calculeze $A^{2016}$.

Barem de corectare.

(2p) Matricea $A$ se poate scrie $A=I_{3}+B$, unde $B=\left(\begin{array}{lll}1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1\end{array}\right)$.

(1p) Deoarece $B^{k}=3^{k-1} \cdot B$, pentru $k \geq 1$, obţinem

(3p) $A^{n}=\left(I_{3}+B\right)^{n}=I_{3}+\sum_{k=1}^{2016} C_{n}^{k} \cdot B^{k}=I_{3}+\left(\sum_{k=1}^{2016} C_{n}^{k} \cdot 3^{k-1}\right) \cdot B$

$$
=I_{3}+\frac{1}{3}\left(\sum_{k=0}^{2016} C_{n}^{k} \cdot 3^{k}-1\right) \cdot B=I_{3}+\frac{4^{2016}-1}{3} \cdot B
$$

(1p) adică, $A^{2016}=\frac{1}{3} \cdot\left(\begin{array}{lll}4^{2016}+2 & 4^{2016}-1 & 4^{2016}-1 \\ 4^{2016}-1 & 4^{2016}+2 & 4^{2016}-1 \\ 4^{2016}-1 & 4^{2016}-1 & 4^{2016}+2\end{array}\right)$.
