It is still not true that (2) implies (1), even if all the elements of A are strictly positive. In fact, suppose that you add a small positive number

to the off-diagonal elements of A, to get the matrix

with all the entries strictly positive. The possible values for the corresponding matrix

are

Then

and

If

is small enough, then

and

will be close to

and

respectively, so you will still get a counterexample to

I don't know whether this is significant for the applications to economics, but mathematically it seems much more plausible that the converse implication

might be true. At any rate, I think it would not be nearly as easy to concoct a counterexample.