An integral domain is a commutative ring with unity and no zero divisors. The set $M_2(\mathbb{Z})$ of $2\times 2$ integer matrices is not commutative and has zero divisors (e.g. $\begin{pmatrix}1&0\\0&0\end{pmatrix}\begin{pmatrix}0&0\\0&1\end{pmatrix}=0$). Hence it is NOT an integral domain. $\mathbb{Z}$, $\mathbb{Z}_7$ (prime modulus), and $\mathbb{Q}$ are all integral domains.