This result can be obtained by writting the log likelihood of the multivariate gaussian distribution and using the property that for a square matrix $A$ and a vector $x$ of same size: $x^TAx=tr(Axx^T)$. The extra term $-\lambda||X||_1$ does not come from the likelihood and it there to regularize the problem.

This result can be obtained by writing the log likelihood of the multivariate gaussian distribution and using the property that, for a square matrix $A$ and a vector $x$ of same size, $x^TAx=tr(Axx^T)$. The extra term $-\lambda||X||_1$ does not come from the likelihood, it is there to regularize the problem.