The question is as follows:
I thought of the values on the first row and column as x- and y-vector coordinates.
One thing that I do notice in vector M, is that the sum of each of the two columns is 1. Is that something that is responsible for the collinearity?
Any help will be greatly appreciated.
Think about what it means for the heads of the vectors $v, Mv, M^{2}v, \dots$ to be collinear: they are all on some line. What line is that? In particular, can you describe the line in a form that will help you solve the problem?
Furthermore, about your observation that the columns sum to one: I would also recommend re-expressing this in a more convenient form. In fact, if you can state the two things in similar terms, I suspect you'll be very close to a solution.
Now, the question is what form. Personally, I would put them in typical linear algebra terms, but since the tags include algebra-precalculus, I'll point out that it can be expressed in equations in $x$ and $y$.
(I admit that this is vague. I'll clarify it if it proves unhelpful.)