Product of Row Sum Unity Matrices

Theorem
Let $$\mathbf{A} = \left[{a}\right]_{m n}$$ be an $m \times n$ matrix.

Let $$\mathbf{B} = \left[{b}\right]_{n p}$$ be an $n \times p$ matrix.

Let the row sum of $$\mathbf A$$ and $$\mathbf B$$ be matrices equal $$1$$.

Then the row sum of their (conventional) product is also $$1$$.

Proof
We have that:


 * $$\sum_{i=1}^n a_{ij} = \sum_{i=1}^n b_{ij}=1 \ $$,

Then:

$$ $$ $$ $$ $$