Let \(R\) be a ring with identity element 1. Various criteria are known for \(R\) to be a full \(n\) by \(n\) matrix ring, and the present paper follows on part I by \textit{J. C. Robson} [Commun. Algebra 19, No. 7, 2113-2124 (1991; Zbl 0731.16018)] who showed, for instance, that \(R\) is an \(n\) by \(n\) matrix ring if and only if \(R\) has elements \(a\) and \(f\) such that \(f^n=0\) and \(af^{n-1}+faf^{n-2}+\dots+f^{n-2}af+f^{n-1}a=1\). Several new criteria are given, which involve relations with fewer terms or fewer elements than previous ones. In terms of ``three element relations'', it is shown that \(R\) being an \(n\) by \(n\) matrix ring is equivalent to each of the following: (1) \(R\) has elements \(a\), \(b\), \(f\) such that \(f^n=0\) and \(af^{n-1}+fb=1\); (2) \(R\) has elements \(a\), \(b\), \(f\) such that \(f^n=0\) and \(af^u+f^vb=1\) for some positive integers \(u\) and \(v\) with \(u+v=n\). Also, if \(R\) is the \(k\)-algebra freely generated by elements \(a\), \(b\), \(f\) subject only to the relations in (1), then \(R\) is isomorphic to the \(n\) by \(n\) matrix ring over the free \(k\)-algebra on \(n^2\) generators. The situation for ``two element relations'' is more complicated. For instance if \(n\) is at least 3 then there is no two-element version of (1), i.e. there is no non-trivial ring with elements \(a\), \(f\) such that \(f^n=0\) and \(af^{n-1}+fa=1\). There is a similar problem with the two-element version of (2) when \(u\neq v\). On the other hand, several positive results are proved involving two-element criteria. As an application, the paper ends by showing that certain factor rings of rings of differential operators in characteristic \(p\) are \(p^n\) by \(p^n\) matrix rings.