I think you're correct. But I've never applied the MLE to a linear model...
You may be interested by this part of a Wikipedia article : Linear model - Wikipedia, the free encyclopedia, which seems to confirm your result.
The only difference is that you need the X in the article to be of full rank (for a nxm matrix, "full rank" means that its rank is min(n,m)). But here, I'm quite unsure what stands for X