Question

If A =$\left[\begin{array}{c}i\\ 0\end{array}\begin{array}{c}0\\ -1\end{array}\right]$, than check whether: ${\text{A}}^2=-\text{I,(}{\text{i}}^2=-1)$

• A. True
• B. False

Answer 1

The correct option is B False

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

Taking LHS

$\Rightarrow A^2=A.A$

$\left[\begin{array}{cc}i& 0\\ 0& -1\end{array}\right]\left[\begin{array}{cc}i& 0\\ 0& -1\end{array}\right]$ (using matrix multiplication)

$\Rightarrow \left[\begin{array}{cc}{i}^2+0& 0+0\\ 0+0& 0+1\end{array}\right]$

$\Rightarrow \left[\begin{array}{cc}-1& 0\\ 0& -1\end{array}\right]$ (wihch is not equal to $-I$)

as $I=\left[\begin{array}{cc}-1& 0\\ 0& -1\end{array}\right]$

$\therefore A^2=-I$ is not a valid relation.