Explanation

(a)
For involuntary matrix,
$A^{2} = I$
$\Rightarrow \ \left| A^{2} \right| = |I| \Rightarrow |A|^{2} \Rightarrow |A| = \pm 1$
For idempotent matrix,
$A^{2} = A$
$\Rightarrow \left| A^{2} \right| = |A| \Rightarrow |A|^{2} \Rightarrow |A| \Rightarrow |A| = 0\$or$\ 1$
For orthogonal matrix,
$AA^{T} = I$
$\Rightarrow \ \left| AA^{T} \right| = |I| \Rightarrow |A|\left| A^{T} \right| = 1 \Rightarrow |A|^{2} = 1 \Rightarrow |A| = \pm 1$.
Thus if matrix A is idempotent it may not be invertible.