= Solution
{c}
Use the shape-rate convention for the <Gamma distribution>. Define
$$
B=A^TA,\qquad u_* = B^{-1}A^Tm,\qquad
r=m-Au_*,\qquad
a=\alpha+\frac k2,\qquad
b(u)=\beta+\frac12\|m-Au\|^2.
$$
The zero <null space> assumption makes $B$ a <positive-definite matrix>, and $u_*$ is the unique <ordinary least squares> solution. The <Gaussian likelihood> contributes $\gamma^{k/2}$, which must be retained because $\gamma$ is unknown. Multiplying it by the <improper prior> and the <Gamma distribution> gives the <normal-gamma posterior with a flat prior>:
$$
\boxed{\pi^m(u,\gamma)=\frac1{\mathcal Z}\gamma^{a-1}e^{-\gamma b(u)},\qquad u\in\mathbb R^d,\quad\gamma>0.}
$$
Here the constant density of the <improper prior> cancels on normalization. Put
$$
b_0=\beta+\tfrac12\|r\|^2>0,\qquad
s=a-\tfrac d2=\alpha+\tfrac{k-d}{2}>0.
$$
The <normal equation> $A^Tr=0$ makes the residual orthogonal to the range of $A$. The <orthogonal decomposition by a closed subspace> into that range and its <orthogonal complement> gives
$$
b(u)=b_0+\tfrac12(u-u_*)^TB(u-u_*).
$$
Integration over $u$ by the <Gaussian integral>, followed by integration over $\gamma$ using the <Gamma distribution>, yields
$$
\mathcal Z=\frac{(2\pi)^{d/2}}{\sqrt{\det B}}\frac{\Gamma(s)}{b_0^s}<\infty.
$$
Thus the <posterior distribution> is proper, despite the <improper prior>; here $k\geq d$ follows from the zero <null space> assumption.
<Completing the square> in $u$ and collecting the powers of $\gamma$ give the two <conditional distributions>:
$$
\boxed{u\mid\gamma,m\sim\mathcal N(u_*,\gamma^{-1}B^{-1}),\qquad
\gamma\mid u,m\sim\operatorname{Gamma}(a,b(u)).}
$$
Explicitly their <probability density functions> are
$$
\pi(u\mid\gamma,m)=\frac{\gamma^{d/2}\sqrt{\det B}}{(2\pi)^{d/2}}
e^{-\frac\gamma2(u-u_*)^TB(u-u_*)},\qquad
\pi(\gamma\mid u,m)=\frac{b(u)^a}{\Gamma(a)}\gamma^{a-1}e^{-b(u)\gamma}.
$$
For a joint <maximum a posteriori estimate> relative to $du\,d\gamma$, first fix $\gamma>0$. The unique maximizing $u$ is $u_*$. The remaining log density is $(a-1)\log\gamma-b_0\gamma$ plus a constant. Consequently, if $a>1$,
$$
\boxed{\widehat u_{\mathrm{MAP}}=u_*,\qquad
\widehat\gamma_{\mathrm{MAP}}=\frac{\alpha+k/2-1}{\beta+\|r\|^2/2}.}
$$
The stated assumptions do not always imply $a>1$. If $a=1$, the supremum occurs as $\gamma\downarrow0$ but is not attained on $\gamma>0$; if $a<1$, the density is unbounded there. In either case there is no joint <maximum a posteriori estimate> in the stated parameter space. Calling zero a boundary mode does not make it an admissible positive precision.
If “MAP estimators” instead means separate modes of the <marginal distributions>, integration gives
$$
\pi(u\mid m)\propto\left[b_0+\tfrac12(u-u_*)^TB(u-u_*)\right]^{-a},\qquad
\gamma\mid m\sim\operatorname{Gamma}(s,b_0).
$$
Thus the marginal <maximum a posteriori estimate> of $u$ is always $u_*$, and that of $\gamma$ is $(s-1)/b_0$ when $s>1$, with the same nonattainment at zero when $s\leq1$. Joint and marginal maximization are different operations; the usual joint interpretation gives the preceding boxed pair.
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