Chapter 3.3, Problem 15E

Prove that each of the following subsets $H$ of $GL(2,R)$ is subgroup of the group $GL(2,R)$, the general linear group of order $2$

over $R$.

a. $H=\left\{\left[\begin{array}{cc}a& -b\\ b& a\end{array}\right]|a^2+b^2\ne 0\right\}$

b. $H=\left\{\left[\begin{array}{cc}1& a\\ 0& b\end{array}\right]|b\ne 0\right\}$

c. $H=\left\{\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]|a+c=1,b+d=1,andad-bc\ne 0\right\}$

d. $H=\left\{\left[\begin{array}{cc}a& 0\\ 0& b\end{array}\right]|a\ne 0,b\ne 0\right\}$