*Def.[1]:*

A non-negative $n \times n$ matrix $A$ is called a *non-primitive* if there is no an integer $k$ such that all entries of $A^k$ are positive.[1]

*Def.[2]:*

Let ${\bf A}=(a_{i,j})$ and ${\bf B}=(b_{i,j})$ be two $n \times n$ non-negative matrix over $\mathbb{R}$. Let the positions of zero entries of ${\bf A}$ and ${\bf B}$ be the same ( if $a_{i,j}=0$ then $b_{i,j}=0$ and vice versa ). Then, we say all zero positions of ${\bf A}$ and ${\bf B}$ are in common.

Let ${\bf A}_1,{\bf A}_2, \cdots, {\bf A}_k$ be $n \times n$ non-primitive matrices such that all zero positions of ${\bf A}_i$'s are in common. Let ${\bf B}=\prod_{i=1}^k{\bf A}_i.$

**My question:**

`How to show that the matrix B is a non-primitive matrix.`