Explanation

(b)
$\left( A^{'} - \frac{1}{2}I \right)\left( A - \frac{1}{2}I \right)$=I and $\left( A^{'} + \frac{1}{2}I \right)\left( A + \frac{1}{2}I \right)$=I
$\Rightarrow A + A^{'} = 0\ \ \ \ \ ($subracting the two results$)$
$\Rightarrow A^{'} = - A$
$\Rightarrow A^{2} = - \frac{3}{4}I$
$\Rightarrow \left( \frac{- 3}{4} \right)^{n} = \left( \det(A) \right)^{2}$
$\Rightarrow n$ is even