I was reading about Poisson manifolds and came upon the fact, that the rank of the Poisson matrix $X:=(\{x_i,x_j\})_{1\le i,j \le n}$ at a point $m$ equals the rank of the Poisson structure at that point $m$. So my question is, why that is. I tried to derive it but couldn't see it.
Edit: I came upon this question when I tried to find out why the rank of a poisson structure is always even.
I use slightly different notation (which is more standard). We let $(X,\pi)$ be a Poisson manifold. That is, $X$ is a smooth manifold and $\pi\in\Gamma(X,\wedge^2TX)$ is a bivector field, such that $[\pi,\pi]=0$. The latter is the integrability condition coming from the Schouten bracket which is no doubt covered in the book that you are reading. The Poisson matrix is defined by $\Pi_{ij}=\{x_i,x_j\}$ where $x_1,\dots,x_n$ are local coordinate functions on $X$.
The Poisson structure defined a map $\pi^\sharp:T^*X\to TX$ by $\alpha\mapsto\iota_\alpha\pi:=\pi(\alpha,\cdot)$. The rank of $\pi$ at a point $x\in X$ is defined to be the rank of $\pi^\sharp_x:T_x^*X\to T_xX$. In the local coordinates $x_1,\dots,x_n$, we may write any element $\alpha\in T_x^*X$ as $\alpha=\sum\alpha^idx_i|_x$ and write the Poisson vector as $\pi_x=\sum\pi^{ij}(x)\partial_i\wedge\partial_j$. Now compute the map $\pi^\sharp_x$ in coordinates, and relate it to the Poisson matrix $\Pi$.