His KB article may help you: http://support.apple.com/kb/HT4627?viewlocale=en_US&locale=en_US
In short with 16 devices not all of them can use the same iTunes Match account. Keep in mind that "associating" is not the same as "authorization." You authorize a computer to play protected content and there can be a maximum of 5 computers authorized for one Apple ID. But iOS devices (iPad, iPhone, AppleTV) do not require authorization. They are associated.
iTunes Match is designed to be a single-user service that is never signed out of. It not really designed to be shared. For a device to use an iTunes Match subscription it MUST be signed into the iTunes Store Apple ID which was used to purchase the service. Once a device is authorized and iTunes Match is enabled that device is locked out of switching Apple IDs form90'days.
Since your family has more than one iTunes Store Apple ID you might want to consider purchasing the service for both IDs, on different computers.
Michael's answer is correct, but I wanted to highlight that the one AppleID *per device* rule applies to computers as well; that is, you cannot run Match (or re-download music at all) for more than one AppleID per computer (not per computer account). This is a confusing rule that most people only become aware of when they get burned by it.
It's clear that the "authorization" and "association" rules were invented by different departments at Apple that never spoke to each other.