"I am doing exactly this with a matrix of environmental variables whose sum of rows are all equal to the same value (in this case, 1)."
I'm wondering if this is your problem right here. There is something tickling the back of my brain, although I am not very sure about it. Basically, if they all sum to 1, then the last column can always be calculated from the other columns. An alternate way to consider co-linearity is that a co-linear variable adds no new information, which would also be the case here.
No idea if this is, for sure, the actual answer, but I hope someone else can chime in, or this sets you on a hunt for the right answer.
Cheers.