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Given a matrix $A\in GL_n(\mathbb{Q})$. Can it be expressed as a product of two matrices $B,C$ with $B\in GL_n(\mathbb{Z}[1/p])$ and $C\in GL_n(\mathbb{Z}{(p)})$ GL_{n}(\mathbb{Z}_{(p)})$, where$ \mathbb{Z}{(p)}$\mathbb{Z_p}$ denotes the localization of $\mathbb{Z}$ at $(p)$ ?
# decompositions of matrices over $\mathbb{Q}$
Given a matrix $A\in GL_n(\mathbb{Q})$. Can it be expressed as a product of two matrices $B,C$ with $B\in GL_n(\mathbb{Z}[1/p])$ and $C\in GL_n(\mathbb{Z}{(p)})$ , where $\mathbb{Z}{(p)}$ denotes the localization of $\mathbb{Z}$ at $(p)$ ?