Gaussian integral over Grassmann numbers

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?