Chapter 3.6, Problem 9E

Let $G$ be the additive group $M_2(ℝ)$ of $2\times 2$ matrices over $ℝ$ and $G\text{'}$ the additive group $ℝ$ of real numbers. Define $\varphi :G\to G\text{'}$ by

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

(This mapping is called the trace of the matrix.) Prove or disprove that $\varphi$ is a homomorphism. If $\varphi$ is a homomorphism, find ker $\varphi$ and decide whether $\varphi$ is an epimorphism or a monomorphism.