Problem 1051. Exact binary matrix factorization

Given a binary-valued (only ones and zeroes) MxN matrix C, and K (where K<M and K<N) find a binary-valued MxK matrix A and a binary-valued KxN matrix B such that A*B equals C. e.g. for K = 2, and

C =

       1     1     1     0
       0     0     0     0
       0     1     0     1

then one possible solution is

A =

     0     1
     0     0
     1     0

B =

       0     1     0     1
       1     1     1     0

For this problem, assume that at least one solution exists.

Note that you cannot do matrix multiplication with two logical matrices in MATLAB.