Problem 4. Let the matrix

$$
A=\left(\begin{array}{rcc}
2011 & 2012 & 2013 \\
2013 & 2011 & 0 \\
-2012 & 0 & 2011
\end{array}\right)
$$

Calculate $A^{n}$, where $n \in \mathbb{N}$.
Solution. We observe that $A=2011 \cdot I_{3}+B$, where

$$
B=\left(\begin{array}{ccc}
0 & 2012 & 2013 \\
2013 & 0 & 0 \\
-2012 & 0 & 0
\end{array}\right)=\left(\begin{array}{ccc}
0 & a & a+1 \\
a+1 & 0 & 0 \\
-a & 0 & 0
\end{array}\right)
$$

and

$$
B^{2}=\left(\begin{array}{ccc}
0 & 0 & 0 \\
0 & a(a+1) & (a+1)^{2} \\
0 & -a^{2} & -a(a+1)
\end{array}\right), \quad B^{3}=O_{3}
$$

Thus, $A^{n}=\left(2011 \cdot I_{3}+B\right)^{n}=2011^{n} \cdot I_{3}+n \cdot 2011^{n-1} B+\frac{n(n-1)}{2} \cdot 2011^{n-2} B^{2}$, where $n \geq 2$.

Ministry of Education, Research, Youth and Sports

Dolj County School Inspectorate

Craiova Branch of SSMR

# Local Stage of the National Mathematics Olympiad Grade XI

Craiova, February 9, 2013