Prove that the maximal entries of a positive definite, symmetric, real matrix are on the diagonal

Question

Prove that the maximal entries of a positive definite, symmetric, real matrix are on the diagonal.
(Algebra by Artin, Edition 2, chapter 8, problem 2.1)

This is an assignment problem so I don't want a complete solution. I'm not really sure what "maximal" means. Can anyone tell me that?

Answer 1

Since the matrix entries are real, "maximal" here just means largest.

I would interpret the question to mean: given a positive definite real matrix $M$, let $m = \max_{i,j} M_{ij}.$ Prove that whenever $M_{ij} = m$, $i=j$.