The problem was originally posed by Molnar in [molnar1974matrix].
Various authors followed up this article with both single solutions and parametric families of solutions:
Guy presents an overview and discusses variants of the problem [guy2004unsolved].
A CSP model is presented in [frisch2003constraints]
Unsolved problems in number theory
Constraints for breaking more row and column symmetries
Principles and Practice of Constraint Programming–CP 2003, 318–332, 2003
A matrix problem
American Mathematical Monthly, 383–384, 1974