Let \(M_ n\) be the space of \(n\times n\)-matrices over an algebraically closed field k, the group \(GL_ n\) of invertible matrices acts on \((M_ n)^ r\) diagonally via conjugation. Only in a few cases is it known that the corresponding field of invariant rational functions is rational [cf. E. Formanek, J. Algebra 62, 304-319 (1980; Zbl 0437.16013)]. The author gives a short elegant proof of the fact that, if one replaces a copy of \(M_ n\) by its subvariety of singular matrices, the field of invariants is rational.