Auinger, Chen, Hu, Luo, and the speaker have recently found a new sufficient condition under which a semigroup is nonfinitely based and have applied this condition to certain important classes of infinite semigroups. In the present talk we demonstrate yet another application; its interesting feature is that it requires the full strength of the main result by Auinger et al. Namely, we prove that the absence of a finite identity basis for the semigroup of all upper triangular real 3x3-matrices with 0s and/or 1s on the main diagonal.