Introduction to Applied Linear Algebra Discussion

In a rectangular array of nonnegative real numbers with r rows and c columns, and with r < c, each column contains at least one positive element.

Prove: There exists a positive element for which the sum of the elements of the intersecting row (its row sum) is larger than the sum of the elements of the intersecting column (its column sum).

Note: This is a task for advanced high school math students. The solution should be worked out and explained in detail, preferably using high school math.