The partition function on the 3-sphere of many supersymmetric Chern–Simons-mattertheories reduces, by localization, to a matrix model. We develop a newmethod to study these models in the M-theory limit, but at all orders in the1/N expansion. The method is based on reformulating the matrix model as thepartition function of an ideal Fermi gas with a non-trivial, one-particle quantumHamiltonian. This new approach leads to a completely elementary derivation of theN3/2 behavior for ABJM theory and quiver Chern–Simons-matter theories. In addition, the full series of1/N corrections to the original matrix integral can be simply determined by a next-to-leadingcalculation in the WKB or semiclassical expansion of the quantum gas, and we show that,for several quiver Chern–Simons-matter theories, it is given by an Airy function. Thisgeneralizes a recent result of Fuji, Hirano and Moriyama for ABJM theory. It turns outthat the semiclassical expansion of the Fermi gas corresponds to a strong couplingexpansion in type IIA theory, and it is dual to the genus expansion. This allows us tocalculate explicitly non-perturbative effects due to D0- and D2-brane instantons in the AdSbackground.