what type of rope also thankk you

What u NEED is a guidebook! You can not be thinking of doing a technical route up the Matterhorn without a guidebook or at the Very least some printed notes & sketches, or u will probably be getting lost & endangering yourself, as well as being really unprepared for what lies ahead.
Having said that, a lot of Alpine routes have abseils of 25-30m in them, so u would either need a 60m-70m single rope (i.e. 10mm diameter roughly), or a pair of half ropes (8mm roughly). l do not know whether this is the case with the Matterhorn, but in general it is a good idea to have this amount of rope.
If you're doing a non-technical route up it ( l do not know that if there is one) then again get a guidebook & see what it says about rope & gear.
As for 'type of rope', hopefully I've covered that above, but u will definitely be wanting a dynamic climbing rope if you're doing anything that needs rope, rather than a static 'abseil' or 'caving' rope, which r not designed to take falls.

To be blunt, it really does not sound like you know what you are doing.

l would suggest hiring a guide to take you up the mountain. They would provide all the equipment and enough training to get you to the top.

Technical climbing ropes r ''dynamic'' ropes - which means they have a certain amount of stretch in them so u do not get hurt when they stop u at the bottom of a fall.

Dynamic climbing ropes pretty much only come in the following lengths: 50m (160ft), 60m (200ft) & 70m (less common). These run from about $150 to $250, depending on what u want.

l like PMI, Sterling, Mammut, & Black Diamond ropes.
http://rei.com/product/767454

Good Luck

since you are only 15, about 10 feet of your Mom is clothesline should do it.