Although thermonuclear (Type Ia) supernovae and neutron star mergers are some of the most important astrophysical events, our understanding of these explosions is vague. I will present abundance measurements of elements across the periodic table (Mg, Fe, Co, Ba, and others) that address the nature of both types of explosions. The measurements are based on Keck/DEIMOS spectroscopy of red giants in dwarf galaxies, which experienced a large number of Type Ia supernovae. The iron-peak elemental abundances strongly suggest that the majority of Type Ia supernovae in dwarf galaxies exploded below the Chandrasekhar mass, i.e., the double-degenerate model or the single-degenerate, double-detonation model. The DEIMOS spectra also reveal that barium comes from the r-process and appears in the dwarf galaxies on a timescale similar to iron (at least 100 Myr). Therefore, the mostly likely origin is not supernovae but neutron star mergers. The evolution of the [Ba/Fe] ratio indicates a neutron star merger rate consistent with results from LIGO.