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 (up to sign) is $$Res(P(t),Q(t)) = 81 Res(pt+r,qt+s)^4 = 81(pq-rs)^4.$$