  For a large $n\times m$ Gaussian matrix, we compute the joint statistics,
including large deviation tails, of generalized and total variance - the scaled
log-determinant $H$ and trace $T$ of the corresponding $n\times n$ covariance
matrix. Using a Coulomb gas technique, we find that the Laplace transform of
their joint distribution $\mathcal{P}_n(h,t)$ decays for large $n,m$ (with
$c=m/n\geq 1$ fixed) as $\hat{\mathcal{P}}_n(s,w)\approx \exp\left(-\beta n^2
J(s,w)\right)$, where $\beta$ is the Dyson index of the ensemble and $J(s,w)$
is a $\beta$-independent large deviation function, which we compute exactly for
any $c$. The corresponding large deviation functions in real space are worked
out and checked with extensive numerical simulations. The results are
complemented with a finite $n,m$ treatment based on the Laguerre-Selberg
integral. The statistics of atypically small log-determinants is shown to be
driven by the split-off of the smallest eigenvalue, leading to an abrupt change
in the large deviation speed.
