I'm trying to evaluate a Gaussian integral over Grassmann numbers but not sure if I've made a mistake.
What I want to evaluate is \begin{equation} \left(\prod^N_i\int d\theta^*_i d\theta_i\right)\theta_k \theta^*_l \theta_m \theta^*_n \exp\left(-\theta^*_i B_{ij}\theta_j\right), \end{equation} where $\theta$ is a complex Grassmann number, and $B$ is an $N\times N$ invertible matrix.
First I expanded the integrand to get \begin{align} &\left(\prod^N_i\int d\theta^*_i d\theta_i\right) \frac{1}{(N-2)!}\epsilon^{ln\mu\ldots\nu}\epsilon^{km\alpha\ldots\beta}B_{\mu\alpha}\cdots B_{\nu\beta}\ \theta_1\theta^*_1\theta_2\theta^*_2\cdots\theta_N\theta^*_N - \textrm{(}l,n\textrm{ interchange)} \\ &= \frac{1}{(N-2)!}(\epsilon^{ln\mu\ldots\nu}\epsilon^{km\alpha\ldots\beta}B_{\mu\alpha}\cdots B_{\nu\beta}) - \frac{1}{(N-2)!}(\epsilon^{nl\mu\ldots\nu}\epsilon^{km\alpha\ldots\beta}B_{\mu\alpha}\cdots B_{\nu\beta}). \end{align} The first term has no $B_{lk}B_{nm}$ and the second term has no $B_{nk}B_{lm}$.
Anyway, it is equal to \begin{align} \frac{\partial^2\det B}{\partial B_{nm}\partial B_{lk}} - \frac{\partial^2\det B}{\partial B_{lm}\partial B_{nk}}, \end{align} where \begin{align} \frac{1}{\det B}\frac{\partial^2\det B}{\partial B_{nm}\partial B_{lk}} &= (B^{-1})_{kl}(B^{-1})_{mn} + \frac{\partial (B^{-1})_{kl}}{\partial B_{nm}} \\ &= (B^{-1})_{kl}(B^{-1})_{mn} - (B^{-1})_{kn}(B^{-1})_{ml} \end{align} Therefore, \begin{align} \frac{\partial^2\det B}{\partial B_{nm}\partial B_{lk}} - \frac{\partial^2\det B}{\partial B_{lm}\partial B_{nk}} = 2\,\det B\ \left[(B^{-1})_{kl}(B^{-1})_{mn} - (B^{-1})_{kn}(B^{-1})_{ml}\right] \end{align}
So, my question is, is it right that the factor $2$ is multiplied? If not, where did I make a mistake?
EDIT) I realized that the additional term with the indices $l$ and $n$ are interchanged is not needed; it is already included in the first term. Therefore, \begin{align} \left(\prod^N_i\int d\theta^*_i d\theta_i\right)\theta_k \theta^*_l \theta_m \theta^*_n \exp\left(-\theta^*_i B_{ij}\theta_j\right) &= \frac{1}{(N-2)!}(\epsilon^{ln\mu\ldots\nu}\epsilon^{km\alpha\ldots\beta}B_{\mu\alpha}\cdots B_{\nu\beta}) \\ &= \det B\ \left[(B^{-1})_{kl}(B^{-1})_{mn} - (B^{-1})_{kn}(B^{-1})_{ml}\right]. \end{align}
