Imagine a "grid" of people: some number of people arranged in a number of rows and columns in a rectangular formation.

We designate person A as the shortest person in the group of the tallest people of each row. We then designate person B as the tallest person in the group of shortest people in each column.

Who is taller, A or B?

(In reply to

Problem with Solution by Gamer)

You are right Gamer, a 1x1 grid does not apply, but it doesn't have to be a 1x1 grid for A and B to be the same person. Consider the following grid of people (numbers are the heights, in an arbitrary unit):

10 11 12 13

14 15 16 17

18 19 20 21

The tallest of each row are 13, 17, 21. A is the shortest, 13.

The shortest of each column are 10, 11, 12, 13. B is the tallest, lucky number 13.

There is an infinite number of grids this works for.