## PROBLEM 1

Let $X \in M_{3}(C), \quad X=\left(\begin{array}{lll}a & b & c \\ 0 & a & b \\ 0 & 0 & a\end{array}\right)$. Calculate $X^{n}, n \in N$.
Solution:

$X$ decomposes as $X=A+B$, where $A=\left(\begin{array}{lll}a & 0 & 0 \\ 0 & a & 0 \\ 0 & 0 & a\end{array}\right)=a I_{3}$ and $B=\left(\begin{array}{lll}0 & b & c \\ 0 & 0 & b \\ 0 & 0 & 0\end{array}\right)$ It is observed that

$A \cdot B=a I_{3} \cdot B=B \cdot a I_{3}=B \cdot A$

1 point

$X^{n}=(A+B)^{n}=C_{n}^{0} A^{n}+C_{n}^{1} A^{n-1} B+\ldots \ldots .+C_{n}^{n} B^{n}$ 1 point

$A=a I_{3}, A^{2}=a^{2} I_{3}, \ldots \ldots ., A^{n}=a^{n} I_{3}, \forall n \in N$ 1 point

$B^{2}=\left(\begin{array}{lll}0 & 0 & b \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{array}\right), B^{3}=\left(\begin{array}{lll}0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{array}\right)=O_{3} \Rightarrow B^{n}=\left(\begin{array}{lll}0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{array}\right), \forall n \in N \quad n \geq 3$

Thus $X^{n}=C_{n}^{0} A^{n}+C_{n}^{1} A^{n-1} B+C_{n}^{2} A^{n-2} B^{2}+O_{3}, X^{n}=a^{n} I_{3}+n \cdot a^{n-1} \cdot B+\frac{n(n-1)}{2} a^{n-2} B^{2}$ 1 point

After substitution, we obtain $X^{n}=\left(\begin{array}{ccc}a^{n} & n a^{n-1} b & n a^{n-1} c+\frac{n(n-1)}{2} a^{n-2} b \\ 0 & a^{n} & n a^{n-1} \\ 0 & 0 & a^{n}\end{array}\right) \ldots \ldots .2$ points