Question

If $A=\left[\begin{array}{cc}i& 0\\ 0& -i\end{array}\right]$ show that $A^2=-I_2$

Answer 1

Given,

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

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

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

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

or, $A^2=-I_2$.

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