Chapter 6.2, Problem 17E

Define $\theta :{{\rm M}}_2\left(\mathbb{Z}\right)\to {{\rm M}}_2\left({\mathbb{Z}}_2\right)$ by

$\theta \left(\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]\right)=\left[\begin{array}{cc}\left[a\right]& \left[b\right]\\ \left[c\right]& \left[d\right]\end{array}\right].$

Prove that $\theta$ is a homomorphism, and describe ker $\theta$.