Duality and integrability of a supermatrix model with an external source

We study the Hermitian supermatrix model involving an external source. We derive the determi-nantal formula for the supermatrix partition function, and also for the expectation value of the characteristic polynomial ratio, which yields the duality between the characteristic polynomial and the external source with an arbitrary matrix potential function. We also show that the supermatrix integral satisfies the 1- and 2D Toda lattice equations as well as the ordinary matrix model.