Use Grassman integration

with \(\eta = \left(\eta_1, \eta_2, \cdots, \eta_n\right)\), \(\lambda = \left(\lambda_1, \lambda_2, \cdots, \lambda_n\right)\) and \(A\) an \(n\) by \(n\) Hermitian matrix

to solve \(\int D\left(\eta, \bar{\eta}\right) \, \exp{\left[-\bar{\eta}^{T} A_{\eta}\right]}\)

 * 1) Diagonalise \(A_{\eta}\): \(I = \int D\left(\eta, \bar{\eta}\right) \, \exp{\left[-\bar{\eta}^{T} u^{\dagger} A_{\eta} u\right]}\)
 * 2) Choose \(v\) such that \(\bar{v}^Tu^{\dagger} A_{\eta} uv = \bar{\eta}A_{\eta}{\eta}\): \(I = \prod_i \int D\bar{v}_i \, Dv_i \, \exp{\left[-\bar{v}^{T} u^{\dagger} A_{\eta} u v\right]}\)
 * 3) Solve out the \(D\) terms: \(I = \prod_i \int d\bar{v}_i \, dv_i \, \left(1 -\bar{v} A_{i} v\right)\)
 * 4) Solve the integrals: \(I = \prod_i A_i\)
 * 5) Simplify: \(I = \det A\)