Question

Let A denote the matrix $\left[\begin{array}{cc}0& i\\ i& 0\end{array}\right]$, where $i^2=-1$, and let I denote the identity matrix $\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$ Then, $I+A+A^2+\cdots +A^{2010}$ is

• A. $\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]$
• B. $\left[\begin{array}{cc}0& i\\ i& 0\end{array}\right]$
• C. $\left[\begin{array}{cc}1& i\\ i& 1\end{array}\right]$
• D. $\left[\begin{array}{cc}-1& 0\\ 0& -1\end{array}\right]$

Answer 1

Answer: (B)

We have,
$A=\left[\begin{array}{cc}0& i\\ i& 0\end{array}\right]$,
$A^2=\left[\begin{array}{cc}0& i\\ i& 0\end{array}\right]\left[\begin{array}{cc}0& i\\ i& 0\end{array}\right]$
$=\left[\begin{array}{cc}{i}^2& 0\\ 0& i^2\end{array}\right]=\left[\begin{array}{cc}-1& 0\\ 0& -1\end{array}\right]$
$=-\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]=-I$
$A^3=A^2\cdot A=\left[\begin{array}{cc}-1& 0\\ 0& -1\end{array}\right]\left[\begin{array}{cc}0& i\\ i& 0\end{array}\right]$
$=\left[\begin{array}{cc}0& -i\\ -i& 0\end{array}\right]$
$=-\left[\begin{array}{cc}0& i\\ i& 0\end{array}\right]=-A$
and $A^4=A^2\cdot A^2=\left(-I\right)\left(-I\right)=I=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$
$\therefore I+A+A^2+A^3+A^4+A^5+\dots +A^{2008}+A^{2009}+A^{2010}$
$=I+A+A^2+A^3+A^4\left[I+A+A^2+A^3\right]+\dots +A^{2008}\left[I+A+A^2\right]$
$=0+0+\dots +\left[I+A+A^2\right]$
$\left[\because I+A+A^2+A^3=0\right]$
$\therefore I+A+A^2=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]+\left[\begin{array}{cc}0& i\\ i& 0\end{array}\right]+\left[\begin{array}{cc}-1& 0\\ 0& -1\end{array}\right]$
$=\left[\begin{array}{cc}0& i\\ i& 0\end{array}\right]$