Maximization of the two-parameter multivariate normal likelihood avoiding logarithm and differentiation

We consider the classical problem of maximizing the likelihood function of the mean vector and the dispersion matrix given a sample from the family of multivariate Normal distributions. While the solution to the problem is extremely well known, all the existing methods of obtaining the solution are more complicated than they have to be. Assuming that the dispersion matrix is positive definite and the sample size is greater than the dimension of the ambient space, we present a method of obtaining the solution that does not use the natural logarithmic function or differentiation in any form and is otherwise very easily accessible.