Question

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

$\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]$

Let $A=\left[\begin{array}{cc}0& i\\ i& 0\end{array}\right]$

$I=\left[\begin{array}{cc}0& i\\ i& 0\end{array}\right]$

Now

$I+A+A^2+A^3+......+A^{2010}$

$=\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]$

Hence, the option $\left(A\right)$ is correct answer.