Consider the non-component-wise transformation

\[
y_{i} \mapsto \frac{y_{i}^{\lambda}}{\lambda \dot{y}^{\lambda-1}}
\]

where \(\dot{y}\) is the geometric mean of the components of \(y \in \mathbb{R}_{+}^{n}\). Using the result of the previous exercise, show that the modified transformation is invertible \(\mathbb{R}_{+}^{n} \rightarrow \mathbb{R}_{+}^{n}\) with Jacobian \(J=|\lambda|^{-1}\).