A precision measurement of the matrix elements for η→π+π-π0 and η→π0π0π0 decays is performed using a sample of (10087±44)×106 J/ψ decays collected with the BESIII detector. The decay J/ψ→γη is used to select clean samples of 631,686 η→π+π-π0 decays and 272,322 η→π0π0π0 decays. The matrix elements for both channels are in reasonable agreement with previous measurements. The nonzero gX2Y term for the decay mode η→π+π-π0 is confirmed, as reported by the KLOE Collaboration, while the other higher-order terms are found to be insignificant. Dalitz plot asymmetries in the η→π+π-π0 decay are also explored and are found to be consistent with charge conjugation invariance. In addition, a cusp effect is investigated in the η→π0π0π0 decay, and no obvious structure around the π+π- mass threshold is observed.