Jacob matrixWatch this thread
The "ad" term is positive (two negatives) so the factor will cancel.
The trace is the sum of the diagonal elements. If your notes say otherwise ...
may be worth using to check your working.
From a very quick look, starting with the nondimensional equations, I agree with the Jacobian in the notes, but the det and tr are not correct. The correct values are in the wolframalpha link I posted earlier.