$\displaystyle
\begin{bmatrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 1 & 0 & -1 & 0 \\ -2 & 1 & 0 & -1\end{bmatrix}
$
That's A and I have to compute A^2010
I'm given a hint to first compute A^2
If you calculate $\displaystyle A^2$ you get the following matrix:
$\displaystyle \begin{bmatrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1\end{bmatrix}$
As you can see its the $\displaystyle I$ matrix and therefore
$\displaystyle A^3=I*A => A$
So you always get either A matrix or the I matrix depending on your square.
If its an even square you get the I matrix otherwise A matrix.
A^2010 you do the math.