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Let $k = \mathbb{Q}(\sqrt{\alpha})$ be a real quadratic number field, where$\alpha$ is a positive square-free integer. Let $\mathcal{O}_k$ be the ring ofintegers of $k$. In this paper, we prove that a certain set of $2 \times 2$singular matrices with entries in $\mathcal{O}_k$ can be written as a productof a bounded number of idempotent matrices. Our main theorem can be viewed as ageneralization of the recent result by Cossu and Zanardo, which studies finitegeneration of certain singular matrices by idempotent matrices over$\mathcal{O}_k$ instead of bounded generation of certain singular matrices byidempotent matrices over $\mathcal{O}_k$ as considered in this paper.

Effectively bounded idempotent generation of certain $2 \times 2$ singular matrices by idempotent matrices over real quadratic number rings