I'm going over the exercise 24.3 here, where the author asks about (what I understand to be) the product of a Jacobian matrix and its transposed inverse, which I think is $$\begin{pmatrix} \frac{\partial x}{\partial s} & \frac{\partial x}{\partial t} \\ \frac{\partial y}{\partial s} & \frac{\partial y}{\partial t} \end{pmatrix} \begin{pmatrix} \frac{\partial s}{\partial x} & \frac{\partial t}{\partial x} \\ \frac{\partial s}{\partial y} & \frac{\partial t}{\partial y} \end{pmatrix}=\begin{pmatrix} \frac{\partial x}{\partial s} \frac{\partial s}{\partial x}+ \frac{\partial x}{\partial t} \frac{\partial s}{\partial y} & \frac{\partial x}{\partial s} \frac{\partial t}{\partial x}+\frac{\partial x}{\partial t} \frac{\partial t}{\partial y} \\ \frac{\partial y}{\partial s} \frac{\partial s}{\partial x}+\frac{\partial y}{\partial t} \frac{\partial s}{\partial y} & \frac{\partial y}{\partial s} \frac{\partial t}{\partial x}+\frac{\partial y}{\partial t} \frac{\partial t}{\partial y} \end{pmatrix} $$
I couldn't see how this matrix is in any way significant. However, if the transpose is removed from the second factor, I believe it reduces to the identity.
Has the author made a mistake here by asking to take the transpose? or does this matrix indeed have some significance as is?
Thank you!