I have a $2\times2$ matrix $$ M= \begin{pmatrix} A & -1\\ 1 & B \\ \end{pmatrix}. $$
Here $A$ and $B$ are skew matrix, the matrix dimension is $L$. Is there a quick way to calculate the Pfaffian of matrix $M$? I want to avoid $2L\times 2L$, only deal with $L \times L$ matrix.
For example:
If I use $$ \mathrm{Pf}(M)^2 =\mathrm{Det}(M)$$ and $$\mathrm{Det}(M) = \mathrm{Det}(1+AB),$$ I only deal with $L\times L$ matrix. However, I lose the sign of $\mathrm{Pf}(M)$, since it is $\pm \sqrt{\mathrm{Det}(M)}$.
Do you have any suggestion? Is $ \mathrm{Pf}(M)= \mathrm{Pf}(1+AB) $ ?