Lagrangian invariant under left and right multiplication by unitary matrices, slick way to see?

Question

Is there a slick way to see that the Lagrangian$$\mathcal{L} = \text{Tr}(\partial^\mu G\partial_\mu G^{-1}),$$where $G$ is an $N \times N$ unitary matrix, is invariant under left and right multiplication by unitary matrices?

Answer 1

Is that what you are looking for?

$$\begin{split} \text{Tr}(\partial^\mu(A G)\partial_\mu (AG)^{-1}) &= \text{Tr}(A\partial^\mu G\partial_\mu G^{-1}A^{-1})\\ &= \text{Tr}(\partial_\mu G^{-1}A^{-1}A\partial^\mu G) \\ &= \text{Tr}(\partial_\mu G^{-1}\partial^\mu G) \\ &= \text{Tr}(\partial^\mu G \partial_\mu G^{-1}) \end{split}$$