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Let $A$ denote the matrix $\left[\begin{array}{ll}0 & i \\ i & 0\end{array}\right]$, where $i^2=-1$, and let $I$ denote the identity matrix $\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]$. Then, $I+A+A^2+\ldots+A^{2010}$ is
$\left[\begin{array}{ll}0 & 0 \\ 0 & 0\end{array}\right]$
$\left[\begin{array}{ll}0 & i \\ i & 0\end{array}\right]$
$\left[\begin{array}{ll}1 & i \\ i & 1\end{array}\right]$
$\left[\begin{array}{cc}-1 & 0 \\ 0 & -1\end{array}\right]$
Solution
(b)
We have,
$A=\left[\begin{array}{ll}0 & i \\ i & 0\end{array}\right]$
$A^2=\left[\begin{array}{ll}0 & i \\ i & 0\end{array}\right]\left[\begin{array}{ll}0 & i \\ i & 0\end{array}\right]$
$=\left[\begin{array}{cc}i^2 & 0 \\ 0 & i^2\end{array}\right]=\left[\begin{array}{cc}-1 & 0 \\ 0 & -1\end{array}\right]$
$=-\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]=-I$
$A^3=A^2 \cdot A=\left[\begin{array}{cc}-1 & 0 \\ 0 & -1\end{array}\right]\left[\begin{array}{ll}0 & i \\ i & 0\end{array}\right]$
$=\left[\begin{array}{cc}0 & -i \\ -i & 0\end{array}\right]$
$=-\left[\begin{array}{ll}0 & i \\ i & 0\end{array}\right]=-A$
and $A^4=A^2, A^2=(-I)(-I)=I=\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]$
$\therefore I+A+A^2+A^3+A^4+A^5$
$\quad+\ldots+A^{2009}+A^{2009}+A^{2010}$
$=I+A+A^2+A^3+A^4\left[I+A+A^2+A^3\right]$
$+\ldots+A^{2005}\left[I+A+A^2\right]$
$=0+0+\ldots+\left[I+A+A^2\right]$
$=\left[\begin{array}{ll}0 & i \\ i & 0\end{array}\right]$
