## SUBJECT I

Consider the matrix $A \in M_{2}(\square), A=\left(\begin{array}{ll}3 & 3 \\ 2 & 2\end{array}\right)$.

a) Calculate $A^{n}, n \in \square^{*}$.

b) Solve the equation $X^{2015}=A, X \in M_{2}(\square)$.
## SUBJECT I

a) $A^{2}=5 A(1 \text{p})$

$A^{n}=5^{n-1} \cdot A(0.5 \text{p})$

Completion by induction $(0.5 \text{p})$
b) $\operatorname{det} X=0(1 \text{p})$

$$
\begin{aligned}
& X^{2}=t \cdot X, t \in \square, t=\operatorname{Tr}(X)(1 \text{p}) \\
& X^{2015}=t^{2014} \cdot X \Rightarrow X=\frac{1}{t^{2014}} \cdot A(1 \text{p}) \\
& X^{2015}=\frac{1}{t^{2014 \cdot 2015}} \cdot 5^{2014} \cdot A \Rightarrow t^{2014}=\frac{1}{\sqrt[2015]{5^{2014}}}(1 \text{p}) \\
& X=\frac{1}{\sqrt[2015]{5^{2014}}} \cdot A(1 \text{p})
\end{aligned}
$$