Hello. I am a biologist trying to understand a certain proof in a paper (link included to paper). It deals with showing that the matrix form of a Lyapunov function's derivative is positive definite.

The pic related shows the matrix. I do not understand proposition (ii). The determinant for a symmetric positive definite matrix should be greater than zero, how are they stating it is less than zero?

I understand proposition (i), because if g(x,y) <0 then -g(x,y) > 0, which is the determinant of the upper-left submatrix (Slyvester's criterion).

I have been stuck on this for two days, no one is answering me anywhere on the internet.

Please help me out!

Here is a link to the entire paper
https://reader.elsevier.com/reader/sd/pii/S0893965903900966?token=C079C40F445A1C320CB768EA7A380C31A9D096D3E36B0F3863A1D55B81B51ABDB8279A4EEEAFF42EA92A7186875C9C9D&originRegion=us-east-1&originCreation=20210416044616