The Jacobian matrix consists of coefficients of $t^3,t^2u,tu^2,u^3$ in the following $4$ partial derivatives $$\partial_p F(t,u,pt+ru,qt+su) = \partial_{y}F(t,u,pt+ru,qt+su)t,\\ \ldots\\ \partial_s F(t,u,pt+ru,qt+su)= \partial_{z}F(t,u,pt+ru,qt+su)u.$$
Following B. Wellington's comment, (up to permutation of columns) this matrix is by definition the Sylvester matrix of the dehomogenizations $P(t)$ and $Q(t)$ of $P(t,u) = \partial_{y}F(t,u,pt+ru,qt+su)$ and $Q(t,u) = \partial_{z}F(t,u,pt+ru,qt+su)$ respectively, regarded as homogenous polynomials in two variables $t$ and $u$.
In our particular case, $P(t,u) = 3(pt + ru)^2$ and $Q(t,u) = 3(qt+su)^2$. Since the resultant of two polynomials is defined as the product of the differences of their roots, it follows immediately that the determinant of the Jacobian matrix is $$Res(P(t),Q(t)) = 81 Res(pt+r,qt+s)^4 = 81(pq-rs)^4.$$