TITLE: $M_n(\mathbb{Z}) = SL_n(\mathbb{Z}) + SL_n(\mathbb{Z})$
QUESTION [6 upvotes]: I came across the following problem and I'm stuck.

Let $n>1$ be an even integer, and $A\in \mathcal{M}_n(\mathbb{Z})$.
Show that there exist $B,C \in \mathrm{SL}_n(\mathbb{Z})$, such that  $A=B+C$.

Any hint is appreciated.

REPLY [2 votes]: Ideas :
Use Smith normal form. Then $M=PDQ$ where $D$ is a diagonal matrix and $P,Q$ are invertible matrices in $\mathcal{M}_n(\mathbb{Z})$ with $det(P)=det(Q)=\pm{1}$.