Question

If $A=\left[\begin{array}{cc}1& 0\\ 0& -1\end{array}\right],B\left[\begin{array}{cc}0& -1\\ 1& 0\end{array}\right]andc=\left[\begin{array}{cc}0& i\\ i& 0\end{array}\right]$, then $A^2=B^2=C^2=$

• A. $I^2$
• B. $I$
• C. $-I$
• D. `$21$

Answer 1

The correct option is A $I^2$

$A=\left[\begin{array}{cc}1& 0\\ 0& -1\end{array}\right],A^2=\left[\begin{array}{cc}1& 0\\ 0& -1\end{array}\right]\left[\begin{array}{cc}1& 0\\ 0& -1\end{array}\right]$

$=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]=I^2$ ........ $\left(1\right)$

$B=\left[\begin{array}{cc}0& -1\\ 1& 0\end{array}\right],B^2=\left[\begin{array}{cc}0& -1\\ 1& 0\end{array}\right]\left[\begin{array}{cc}0& -1\\ 1& 0\end{array}\right]$

$=\left[\begin{array}{cc}-1& 0\\ 0& -1\end{array}\right]\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]=I^2$...... $\left(2\right)$

$C=\left[\begin{array}{cc}0& i\\ i& 0\end{array}\right],C^2=\left[\begin{array}{cc}0& i\\ i& 0\end{array}\right]\left[\begin{array}{cc}0& i\\ i& 0\end{array}\right]=\left[\begin{array}{cc}{i}^2& 0\\ 0& i^2\end{array}\right]$

$=\left[\begin{array}{cc}-1& 0\\ 0& -1\end{array}\right]\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]=I^2$ ........ $\left(3\right)$

from $\left(1\right),\left(2\right)$ & $\left(3\right)$

$A^2=B^2=C^2=I^2$

So, $\left(A\right)$ is the right option.