It's true, iTunes Match would bypass your inability to get the computer to recognize the iPad for music syncing. However, I recommend you read how it works carefully before getting it. I like it myself, but it's not a service everyone appreciates. It also costs about $25 a year.
On the plus side, if you like iTunes Radio the subscription removes all the commercials.
It's main features.
It matches, rather than uploads, all the music it can. So if you have a song ripped from a CD, in most cases you won't get that exact song. What you will get is the iTunes Store version. If the song/album can not be matched, then it is uploaded to your iCloud account. There is no apparent difference in how it looks in the Music app; but it does count agains't your iCloud storage limit. Unless you have a lot of music that does not exist in the iTunes Store, this has little impact.
When you play a song in the Music app, it streams by default. This saves space on the iPad. However you do have the option of tapping the cloud icon (for playlist, album, or song) and downloading the song(s) so that they are local on the iPad. As long as the iPad has plenty of free storage, they stay. If the iPad starts to run out of storage space it will remove the less played songs from local storage. You can still stream them.
Another nice things about iTunes Match is that it automatically syncs your playlists, keeping them up to date without the need for manual syncing.
Here's Apple's page about the service.
Apple - iTunes - iTunes Match
I'll also warn you that iTunes Match often fails to work 100% on the first sync/upload from the computer, especially if the library is large. Most people report that it fixes itself over time, or after a few forced re-syncs with the service.
iTunes Match is not the only way to get all your iTunes library on the iPad, but I admit ignorance of the other methods, most of which require third party syncing apps on your computer. I'll leave explaining them to others who use those solutions.
