Summary: We introduce a calculational tool useful in computing ratios and products of characteristic polynomials averaged over Gaussian measures with an external source. The method is based on Dyson’s Brownian motion and Grassmann/complex integration formulas for determinants. The resulting formulas are exact for finite matrix size \(N\) and form integral representations convenient for large \(N\) asymptotics. Quantities obtained by the method are interpreted as averages over standard matrix models. We provide several explicit and novel calculations with special emphasis on the \(\beta =2\) Girko-Ginibre ensembles.