Irreducible representations of the quantum Weyl algebra at roots of unity given by matrices

To describe the representation theory of the quantum Weyl algebra at an $l$th primitive root $\gamma$ of unity, Boyette, Leyk, Plunkett, Sipe, and Talley found all nonsingular irreducible matrix solutions to the equation $yx-\gamma xy=1$, assuming $yx\neq xy$. In this note, we complete their result by finding and classifying, up to equivalence, all irreducible matrix solutions $(X, Y)$, where $X$ is singular.