Follow the directions on the Dragtooth main page to arrive at the toe of the Matterhorn Glacier. This route follows the couloir just west of Matterhorn Peak, and this couloir separates the two formations. I can't actually find a name for this couloir, I'm sure it has one, but it is on the opposite side of Matterhorn Peak from the East Couloir, so it made sense. Well, kind of. I guess from the standpoint of The Dragtooth, it could be the East Couloir as well.
The West Couloir has two branches, and the east branch of the West Couloir is the easier of the two. Both couloirs are extraordinarily loose and unpleasant. The east branch is the least unpleasant of the two,but it was choked with rock-hard water ice even as late into the season as October. Without good, sharp, steel crampons, the alternative is negotiate the rock on the west side of the ice. This rock is loose, loose, loose. Don't pull hard on it, and be careful where you stand. Some sections are not just loose, but sandy and unstable as well.
Once you top out at the top of the couloir, it gets much better. The strongest attribute of this route is the directness. The only more direct route is a 4th class route that wends up the North Face.
From the notch at the top of the couloir, descend 150-200' down the south side, to bypass a difficult notch, then clamber up onto the crest of the ridge and enjoy easy climbing and fresh exposure, well over 1,000 feet of it. There are many small gendarmes along the ridge, but they are easy to either traverse over or around without straying far from the crest.
To spice up your ascent a little more, you could traverse over directly from Matterhorn Peak, keeping to the ridge. In the words of Peter Croft, from The Good, the Great, and the Awesome, "From the summit of Matterhorn traverse easily to the top of the Dragtooth." Sounds easy enough.
The usual for a scramble. Early season, crampons and axe. After that, good shoes and desire. Plenty of water along the route until you leave the glacier.
If you have information about this route that doesn't pertain to any of the other sections, please add it here.