I need to evaluate two Grassmann integrals, one over "real" Grassmann variable another one over complex variables.

Lets start with the real one first :

The prototype we have for $n$ real Grassmann variables :

$$ \tag{1}\int d^n\psi \exp{[\frac{1}{2}\psi^T M\psi]} ~=~ (\det M)^{1/2} $$

Now we can shift the integration variable and use the shift invariance of the integration :

If we replace $\psi\rightarrow \psi - M^{-1}\eta$ Then argument inside exponential becomes :

$$ [\frac{1}{2}\psi^T M\psi] ~\longrightarrow~ \frac{1}{2}(\psi - M^{-1}\eta)^T M(\psi - M^{-1}\eta) $$ $$ =\frac{1}{2}[\psi^T M\psi + \eta^T (M^{-1})^T MM^{-1}\eta \space\underbrace{-\eta^T (M^{-1})^T M\psi - \psi^TMM^{-1}\eta}] $$

$$ =\frac{1}{2}\psi^T M\psi+\frac{1}{2}\underbrace{\eta^T (M^{-1})^T MM^{-1}\eta}_{\text extra}+\eta^T\psi. \tag{2} $$

This "extra" term should vanish, but how? I will run into same problem is do the same Gaussian integral over complex Grassmann variables.