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Lately, I´ve been struggling with math homework and came across a question I´m not sure how to answer. I will be glad for any help...

Suppose we have matrix $A$ (size $n\times n$) and its inverse (lets call it $B$). They are both non-negative in the sense that all their elements $A_{ij}$ and $B_{ij}\geq 0$, where $1\leq i,j\leq n$.

The question is, what can we say about these matrices - everything must be justified.

This where I got so far: 1) $A$ is regular (otherwise it wouldn't have and inverse - I don't think I have to justify this statement)

Are there any other features? I think I can justify some of them by using minor matrices, but I'm not sure how :-(

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Welcome to math.stackexchange! I just improved tex formatting in your post. – Davide Giraudo Oct 6 '12 at 15:41
At least, when $n=2$ we can say that either $A$ is diagonal or has the form $\pmatrix{0&a\\a'&0}$ where $a>0$, $a'>0$. – Davide Giraudo Oct 6 '12 at 15:50
up vote 1 down vote accepted

You know that $AB= I_n$.

Since for all $i \neq j$ you have $\sum_k A_{ik}B_{kj}=0$ it follows that for all $i,j,k$ with $i \neq j$ you have either $A_{ik}=0$ or$B_{kj}=0$.

Now, since $\sum_k A_{ik}B_{ki} \neq 0$, for each $i$ you can find some $k_i$ so that $A_{ik_i} \neq 0$ and $B_{k_i i} \neq 0$. Combining this with the above, you can prove that $B_{k_i j}=0 \forall j\neq i$ and $A_{j k_i}=0$ for all $j \neq i$.

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