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 are non-negative and they sum to 1.

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.