So, the index of an (Walrasian/general equilibrium) equilibrium point is determined as the sign of $(-1)^{L-1} \times \det M$ where $M$ is a matrix and $M_{ij} = \frac{\partial{Z_i}}{\partial {p_j}}$, where $Z_i$ is excess demand for $i$th product and $p_j$ is price of $j$th product, and $L$ is the number of total products.
I get what $\det M$ is doing here, but I cannot get why $(-1)^{L-1}$ is there. Can anyone explain why? If explanations can be done with some analogy to Poincare-Hopf theorem, I would appreciate more.