Let $A$ be an antisymmetric matrix.
Usually, one proves that for a Grassmann integral of the type, $$\int d\psi d\theta \exp( \psi^T A \theta) = \det(A)$$ where $\psi$ and $\theta$ are vectors of Grassmann variables.
I have a different problem which I feel should trivially reduce to the same, but it can't figure it out:
$$\int d\psi d\theta \exp( 1/2 \psi^T A \psi + 1/2 \theta^T A \theta)$$
I think it should be possible to show that my expression is equivalent to the upper one, but the necessary variable substitution escapes me.