Is there a case that 2 positive matrices multiplication returning 0 element?

Question

If two matrices do not have negative number nor zero, I don't think it can return zero element after multiplication as each element is summation of only positive numbers. But I just operates large matrices without zero and negative number, I got some 0 in result. Is there a case that this can happen?

Answer 1

Suppose that $A=(A_{i,j})_{n\times m}$ and $B =(B_{i,j})_{m \times p}$ are two matrices with nonnegative entries; i.e.,$A_{i,j},B_{i,j} > 0$. Now, the multiplication matrix $AB$ has the entries values $$(AB)_{i,j} = \sum_{k=1}^{m} A_{i,k} B_{k,j} > 0$$ So it is impossible to have zeros or negative entries in the multiplication matrix $AB$.