Which of the following maps are homomorphisms? If the map is a homomorphism, what is the kernel? Which of the following maps are homomorphisms? If the map is a homomorphism, what is the kernel?
(a) $\phi : \newcommand{\GL}{\operatorname{GL}}\newcommand{\RR}{\mathbb{R}}$ $\RR^{∗} \to \GL_2(\RR)$ defined by $\phi(a) = \begin{pmatrix}1 & 0\\ 0 & a\end{pmatrix}$
I solved this by $ \phi(a)\phi(b)$= matrices multiplied together to get \phi(ab) which is a hom and the $Ker(\phi)=\phi^{-1}(I_2)= {1}$
(b) $\phi : \RR\to \GL_2(\RR)$ defined by $\phi(a) = \newcommand{\bmat}{\begin{pmatrix}}\newcommand{\emat}{\end{pmatrix}}\bmat 1 & 0\\ a & 1\emat$
I solved this by $ \phi(a)\phi(b)$= matrices multiplied together to get \phi(a+b) which is a hom and the $Ker(\phi)=\phi^{-1}(I_2)= {0}$
(c) $\phi : \GL_2(\RR) \to\RR$ defined by $\phi\left(\bmat a& b\\c &d\emat\right) = a + d$
(d) $\phi : \GL_2(\RR)\to \RR^*$ defined by $\phi\left(\bmat a & b\\ c & d\emat \right) = ad − bc$
(e) $\phi : \newcommand{\MM}{\mathbb{M}}\MM_2(\RR)\to\RR$ defined by $\phi\left(\bmat a & b\\ c & d\emat\right) = b,$
where $\MM_2(\RR)$ is the additive group of 2 × 2 matrices with entries in $\RR$
I am confused how to do c,d,e Can some one point me in the right direction please
 A: You are on the right track. The remaining parts can also be done by the same procedure.
Let
$$
A = \begin{pmatrix}
a_1 & b_1 \\
c_1 & d_1
\end{pmatrix}
\qquad\text{and}\qquad
B = \begin{pmatrix}
a_2 & b_2 \\
c_2 & d_2
\end{pmatrix}.
$$
Part (c): Suppose $A,B \in \mathrm{GL}_2(\mathbb{R})$. Compute $AB$ and then $\phi(AB)$. Check whether or not this is equal to $$\phi(A) + \phi(B) = a_1 + d_1 + a_2 + d_2.$$ Hint: It is not a homomorphism. Also note that $\phi$ is the trace function.
Part(d): Suppose $A,B \in \mathrm{GL}_2(\mathbb{R})$. Use the previous computation of $AB$ to figure out $\phi(AB)$ in this case, and compare it with $$\phi(A) \cdot \phi(B) = (a_1 d_1 - b_1 c_1)(a_2 d_2 - b_2 c_2).$$ Hint: It is a homomorphism. Also note that $\phi$ is the determinant function. The kernel will be the set of all those matrices whose determinant is $1$. (This space is denoted $\mathrm{SL}_2(\mathbb{R})$.)
Part (e): Suppose $A,B \in \mathrm{M}_2(\mathbb{R})$. Compute $A+B$ and then $\phi(A+B)$. Compare this with $$\phi(A) + \phi(B) = b_1 + b_2.$$ Hint: It is a homomorphism. The kernel will be the set of all those matrices that send their $(1,2)$ component to $0$.
