  Within the random matrix theory approach to quantum scattering, we derive the
distribution of the Wigner-Smith time delay matrix $\mathcal{Q}$ for a chaotic
cavity with uniform absorption, coupled via $N$ perfect channels. In the
unitary class $\beta=2$ we obtain a compact expression for the distribution of
the full matrix in terms of a matrix integral. In the other symmetry classes we
derive the joint distribution of the eigenvalues. We show how the large $N$
properties of this distribution can be analysed in terms of two interacting
Coulomb gases living on two different supports. As an application of our
results, we study the statistical properties of the Wigner time delay
$\tau_{\mathrm{W}} = \mathrm{tr}[\mathcal{Q}]/N$ in the presence of absorption.
