Question

If $A=\left(\begin{array}{cc}i& 0\\ 0& i\end{array}\right)$, then $A^{20}$ equals to

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

Answer 1

The correct option is B $\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)$

$A=\left(\begin{array}{cc}i& 0\\ 0& i\end{array}\right)$
$\Rightarrow A^2=i^2\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)$

$A^4=\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)=I$

$\therefore (A^4{)}^5=I^5=I$

$\Rightarrow A^{20}=\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)$