Hint: When a coefficient matrix is singular (non-invertible,).    From Wordnik.com.  [Yahoo! Answers: Latest Questions] Reference

Diophantus generally seems to prefer procedures that lead to invertible rational parametrisations; when his procedure leads to a non-invertible rational parametrisation (a so-called unirrational parametrisation) it is generally the case that one can show nowadays that no invertible rational parametrisation is possible.    From Wordnik.com.  [Citizendium, the Citizens' Compendium - Recent changes [en]] Reference

He would have seen the distinction between his invertible and non-invertible rational parametrisations only in so far as the procedures that give the latter make special assumptions, and (arguably) in so far as non-invertible rational parametrisations miss some rational points (very easily found ones in some of Diophantus's examples).    From Wordnik.com.  [Citizendium, the Citizens' Compendium - Recent changes [en]] Reference