A note on singularity categories and triangular matrix algebras

Let $\mathrm{\Lambda }=\left[\begin{array}{cc}A& 0\\ M& B\end{array}\right]$ be an Artin algebra and ${}_B{M}_A$ a $B$-$A$-bimodule. We prove that there is a triangle equivalence $D_{sg}(\mathrm{\Lambda })\cong D_{sg}(A)\coprod D_{sg}(B)$ between the corresponding singularity categories if ${}_{B}M$ is semi-simple and $M_A$ is projective. As a result, we obtain a new method for describing the singularity categories of certain bounded quiver algebras.