Antimagic square induction problem

Question

I'm currently trying to solve all the problems in Jay Cumming's book of proofs, and on the induction section, I'm having trouble solving a certain problem involving antimagic squares, the exercise is formulated as follows:
An antimagic square is an n by n matrix where each line, row and diagonal sums to a distinct value. Prove that, for every integer n larger than or equal to 2, there exists an n by n antimagic square all of whose entries are positive integers.
My approach to solving this problem was a bottom-up approach, by trying to construct an n+1 by n+1 square starting from the n by n one, however, after some attempts, I still have no argument to guarantee that all the sums are distinct.
I'd appreciate any hints on how to solve the problem, perhaps faults within my approach or an entirely new one altogether. Thanks.

Answer 1

A non-inductive solution . . .

Fill the $n{\times}n$ matrix with $n^2$ distinct powers of $2$.