Question on the method of Gauss-Legendre

Question

The Gauss-Legendre method seems to be one of the most important methods in solving differential equations numerically. I have a few questions related to it:

1. My teachers always assumed - without a proof - that, if $(A,b,c)$ is the Butcher tableau associated to the Gauss-Legendre method, then $A$ is always invertible.

2. It always seems to me that the coefficients $b_{i}$ are non-negative? Why is that so?

3. I have read somewhere that the two facts above can be derived from the formula $$z^*(BA+A^*B)z=|b^*z|^{2},$$ where $z\in\mathbb{C}^{s}$ is an arbitrary vector. Where does this formula come from, and how does it imply the first two points?

Is this all to be found in the Legendre polynomials?