Help with variation of the 3-dimensional $\sigma$-model action Consider the following action
$$S=\int\mathrm{d}^3x\sqrt{h}\left[R^{(3)}-\frac{1}{4}\mathrm{Tr}\left(\chi^{-1}\chi_{,i}\chi^{-1}\chi^{,i}\right)\right]$$
where $h$ is the determinant of the 3-dimensional metric tensor $h_{ij}$ and $R$ is the Ricci scalar.
I want to get the equations of motion
\begin{align*}
\left(\chi^{-1}\chi^{,i}\right)_{;i}&=0,\\
R_{ij}&=\frac{1}{4}\mathrm{Tr}\left(\chi^{-1}\chi_{,i}\chi^{-1}\chi_{,j}\right).
\end{align*}
However, how do I perform the variation on the trace?
 A: Your second equation involving $R_{ij}$ is just Einstein's equation with the right side the energy momentum tensor, so I'll just talk about the first equation which is specific to the principal chiral model (sigma model on a Lie group).
The variation of the trace is proportional to
$$-\mathrm{Tr}\left(\chi^{-1}\chi_{,i}\delta\left(\chi^{-1}\chi^{,i}\right)\right)=\mathrm{Tr}\left(\partial_i\left(\chi^{-1}\chi_{,i}\chi^{-1}\right)\delta\chi-\chi^{-1}\chi_{,i}\delta\left(\chi^{-1}\right)\chi^{,i}\right)$$
where I've integrated by parts in the first term. This can be taken to be a covariant derivative if you include the inhomogeneous term from the derivative acting on $\sqrt{h}$, but I will just take a flat metric for convenience (the more general case follows exactly the same as below).
The key is that from the identity $\chi\chi^{-1}=\chi^{-1}\chi=I$ we have the following identities for the derivative and variations,
$$\partial_i\chi^{-1}=-\chi^{-1}\chi_{,i}\chi^{-1},\qquad \delta\chi^{-1}=-\chi^{-1}\delta\chi\,\chi^{-1}.$$
So using these and the cyclic commutation of matrices inside a trace, the variation of the trace becomes
$$\mathrm{Tr}\left(\partial_i\left(\chi^{-1}\chi_{,i}\right)\chi^{-1}\delta\chi\right)$$
And now since $\chi^{-1}\delta\chi$ is an arbitrary Lie algebra element, setting the variation equal to zero gives the equation of motion
$$\partial_i\left(\chi^{-1}\chi_{,i}\right)=0$$
