Solution
= Solution
Using part (i) and $A^\dagger=-A$ in the <Cayley transform (Lie theory)> with argument $-A$ gives
$$
\begin{aligned}
U^\dagger&=[(I+A)^{-1}]^\dagger(I-A)^\dagger\\
&=(I-A)^{-1}(I+A)\\
&=(I+A)(I-A)^{-1}=U^{-1}.
\end{aligned}
$$
The factors commute because $I-A$ and $I+A$ are polynomials in the same <matrix>, and an invertible commuting matrix also commutes with the other factor's <matrix inverse>. Both inverses exist by part (ii). Thus \b[$U$ is unitary].