Solution
= Solution
{c}
Expand the <Bell state> in the standard basis:
$$
\begin{aligned}
\langle\Omega|A\otimes B|\Omega\rangle
&=\frac12\sum_{j,k=0}^1\langle j|A|k\rangle\langle j|B|k\rangle\\
&=\frac12\sum_{j,k}A_{jk}B_{jk}
=\boxed{\frac12\operatorname{tr}(A^TB).}
\end{aligned}
$$
The transpose is not an adjoint: the formula is complex-linear in both operators. The equality follows since $(A^T)_{kj}=A_{jk}$.