MLE using multivariate normal distribution
The "trick" you are referring to is a property of the trace of a product of matrices, namely

$${\rm tr}(ABC) = {\rm tr}(BCA)$$

assuming the dimensions are conformable of course.  

Now, note that the dimension of $(x_n-\mu)^T\Sigma^{-1} (x_n-\mu)$, for each observation, is $1 \times 1$. So trivially,

$${\rm tr}\Big[(x_n-\mu)^T\Sigma^{-1} (x_n-\mu)\Big] = (x_n-\mu)^T\Sigma^{-1} (x_n-\mu) $$

But since also, from the mentioned property of the trace,

$${\rm tr}\Big[(x_n-\mu)^T\Sigma^{-1} (x_n-\mu)\Big] = {\rm tr}\Big[\Sigma^{-1} (x_n-\mu)(x_n-\mu)^T\Big]$$

combining one gets the alternative expression for the mutlivariate normal log-likelihood.