There isn't really any need to deal with matrices for just geometry. The ability to use matrices however is a great tool. For one I don't know of a more elegant

There isn't really any need to deal with matrices for just geometry. The ability to use matrices however is a great tool. For one I don't know of a more elegant expression for the area of a triangle than just |P Q R| |1 1 1| / 2 Rational trigonometry seems to me like it's just linear algebra, except it uses the squared sine instead of the cosine just to get a notion of angle that is 0 for parallel line. Lot's of linear algebra features are removed, but I'm not seeing much benefit from doing so.