It is interesting to see how the non-intersecting case shows up in this scenario. It is the case where M becomes rank 2 or non-invertible.

what would happen if M is not invertible? I think it would mean that ray and triangle are parallel.

@goku2021, If M was not invertible that would mean that that the columns of M are linear combination of one another so yes - the ray could be parallel to a triangle edge.

I think there are a few conditions to check to make sure the point we find is actually an intersection of the ray and triangle: t >= 0, v >= 0, u >= 0, u + v <= 1.

Once we have u, v, and t, see if u, v and 1-u-v are positive and t is positive, then the ray intersects.