Craft Commerce: Hard code different product prices based on currency?
You can add custom fields to store the 'other' price for your products (and/or variants), like, as you say, a matrix, or probably even a table would do.  The fields can then be edited by admins in the back end.

In your templates, you'd selectively display whichever price the particular market needs - this would just be basic template logic.

Then, you'll also need a simple custom plugin that listens to onPopulateLineItem and checks which market the customer is in, and modifies the lineItem price accordingly (by retrieving the correct price form the appropriate field).  Once the lineItem prices are modified as appropriate, they'll be used for the payments and stored with the order, factored into totals etc.

There is an example of modifying a lineItem in the docs here:
https://craftcommerce.com/docs/events-reference#commerce_lineitems.onpopulatelineitem

Unless you have other requirements not mentioned, for someone with Commerce experience, that would be really be quite simple to implement.

(Based on ~ a years work with Commerce and related plugin writing).