An Identity for a Gaussian Grassmann integral from Wikipedia I found this identity on Wikipedia: 
$$\int\exp\left[\theta^T A\eta+\theta^T J+K^T\eta\right]d\theta d\eta =\det A\exp\left[-K^TA^{-1}J\right],$$
where the integration variables are Grassmann variable, and $A$ is an invertible matrix. Unfortunately the Wikipedia page gives little context, and I'm not clear if $K$ and $J$ are also supposed to be Grassmann variables. 
Does anybody know how to prove this identity? 
The source is: 
https://en.wikipedia.org/wiki/Grassmann_integral#Gaussian_integrals_over_Grassmann_variables
 A: $J$ and $K$ are $n$-dimensional sources, while $\theta$ and $\eta$ are spinor fields.
I can't provide a complete proof right now, but you might want to take a look here for an explicit calculation of the the analogous case for bosonic fields, which is much simpler.
I'll outline a heuristic derivation of the simpler case of only one spinor field with a single source $J$. In that case the relevant integral is
$\int d\theta \exp\left[-\frac{1}{2}\theta^T A\theta+ J^T\theta\right]$
where $\theta$ is understood to be a vector of $n$ Grassmann variables.
Since Grassmann integrals are invariant with respect to translation of the Grassmann integration variable, it would be nice if there was a transformation that would make the second term independent of $\theta$. Such a transformation is given by
$\theta \rightarrow (\theta - A^{-1}J)^T$, 
where we assume that $A$ is anti-symmetric, $A^T = -A$.
Plugging this into the integrand and crunching through the algebra (mostly multiplication and recalling that we also have $J_i\theta_k = - \theta_kJ_i$) yields (up to sign errors)
$\int d\theta \exp\left[-\frac{1}{2}\theta^T A\theta+ J^T\theta\right] = \int d\theta \exp\left[-\frac{1}{2}\theta^T A\theta+ J^TA^{-1}J\right]$.
The second part of the exponent can come outside the integral, while the remaining part can be shown to be equal to the square root of the determinant of $A$. 
If the above seems a bit flaky I would appreciate input from someone better acquainted with Grassmann integrals.
A: Hint: Complete the square:
$$\begin{align}
&\int\!d^n\theta ~d^n\eta\exp\left[\theta^T A\eta+\theta^T J+K^T\eta+K^TA^{-1}J\right] \cr
~=~& \int\!d^n\theta ~d^n\eta\exp\left[(\theta^T+K^TA^{-1})A(\eta+A^{-1}J)\right] \cr
~=~& \int\!d^n\theta ~d^n\eta\exp\left[\theta^TA\eta\right]\cr
~=~&\det A,\end{align}$$ 
where the last/third equality is modulo sign conventions. In the second equality is used that Berezin integration is translation invariant.
