= Solution
Distinguish reflection at the input terminal, reflection at the receiving load, and a returned signal that traverses the free-space link twice. The input <antenna> is initially matched, so its direct terminal reflection is zero.
In the usual weak-coupling, matched-port <two-port network> model for the <antenna> pair, let the forward and reverse <wave amplitude> transfer coefficients be $t_{21}$ and $t_{12}$. <Electromagnetic reciprocity> gives equal <power> transfer factors $|t_{21}|^2=|t_{12}|^2=\mathcal F$, where $\mathcal F=(G_a\lambda/(4\pi d))^2$ from part (a). The receiving-load reflection returns <wave amplitude> $t_{12}\Gamma_Lt_{21}$ to the original input port. The load-mismatch contribution to the returned <power> is therefore
$$
\boxed{P_r^{\rm load}=P_i|\Gamma_L|^2\mathcal F^2
=P_i\left(\frac{Z_t-Z_i}{Z_t+Z_i}\right)^2
\left(\frac{G_a\lambda}{4\pi d}\right)^4.}
$$
For fixed gains and impedances this returned <power> scales as $\lambda^4$: it contains two link traversals. By contrast, the reflected <power> at the receiving load before that return traversal is $P_i\mathcal F|\Gamma_L|^2\propto\lambda^2$. Calling that local reflected <power> the <power> already back in the input line would omit one propagation loss.
The original wording does not specify an antenna scattering model. The boxed result is the load-dependent echo in an ideal matched-port link, with structural scattering and residual input coupling neglected. The actual total input return can include coherent scattering from the receiving antenna even when $\Gamma_L=0$; gains and effective areas alone do not specify that background. If “reflected back” means only the direct reflection at the stated matched transmitting antenna, its value is $0$. These meanings should not be conflated; the $\lambda^4$ result requires the two-pass echo interpretation and its stated model.
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