I am working on the problem above. So far, I have that since M_1 is rank one, the entire product is rank one, so there is one zero eigenvalue. Therefore, we may take the trace of the resulting product to get the value of the other eigenvalue. For the case where lambda is 1, one can prove that [1 1] is always an eigenvector of the product, and the eigenvalue is 1. But I am not sure where to proceed from here.