How to show that product of two matrices $\mathbf{A}$ and $\mathbf{B}$ of convex combinations as $\mathbf{C=A\*B}$ is also a matrix of convex combinations.


Convex combinations: entries of each column of matrix are non-negative and they sum to 1. 



Product of two matrices of convex combinations
Just turn being a convex combination into vector arithmetic: $(1, \ldots, 1) \cdot C = (1, \ldots, 1) \cdot A \cdot B = (1, \ldots, 1) \cdot B = (1, \ldots, 1)$. The non-negativity is clear I suppose.


