The Jacobian is not a real number. For a map between $\mathbb R^n$ to $\mathbb R^m$ it is at each point where it is evaluated a matrix. In your case a square matrix of dimension $2$. And the determinant of the Jacobian is a real number. Not a positive one in general. For your specific map, the determinant is equal to $1$.
What is the Jacobian matrix? What are its applications? What is its physical and geometrical meaning? Can someone please explain with examples?
It could be set up in any arbitrary way, but all you're really asking/doing is changing the matrix's basis. It contains exactly the same information as the Jacobian, and the Jacobian could be recovered by undoing the change of basis (which is a linear isomorphism).
It properly and distinctively defines the Jacobian, gradient, Hessian, derivative, and differential. The distinction between the Jacobian and differential is crucial for the matrix function differentiation process and the identification of the Jacobian (e.g. the first identification table in the book).
I'd expect the Jacobian of a scalar transform to be $\alpha I_N$ which would have determinant $\alpha ^N$. Note that the determinant of $\alpha 1$ tensored with $1$ is zero since each row is identical. The scalar transform is linear and invertible for $\alpha \neq 0$ so it can't have determinant $0$ for all $\alpha$. As for equation (1.39) have you tried working some small examples, say $2 ...
The Jacobian matrix is a tool used to transform between coordinate systems by taking the rate of change of each component of an old basis with respect to each component of a new basis and expressing them as coefficients that make up an old basis.