Subgroup of invertible matrices proof For this pertinent question I will attach both the written text format as instructed as well as my solution and also the question attached as images if that is easier to view.  ( -sin u cos u 0 0) ( 0 0 1 t) (0 0 0 1) Given that u is an element of the real numbers. Show that T is a subgroup of the group of invertible matrices under matrix multiplication. This is a new area that I am delving into so I am in unfamiliar territory any responses would be great. -nomad609

If $\det (t)=1$ then $t$ is invertible and $\det(t^{-1})=1.$ In this Q you must also show that $t^{-1}$ has the same form as $t,$ and that if $t_1,t_2$ have that form then so does their product $t_1t_2.$ Hint: $T$ (as you have written it ) is a function of $u,$ so write $T(u)$ instead of $T.$ Then what is $T(u)T(-u)$ and what is $T(u_1)T(u_2)$?

If $G$ is a group and $\emptyset \
e H\subset G,$ then $H$ is a subgroup of $G$ iff (i)$\;h^{-1}\in H$ whenever $h\in H,$ and (ii) $\;h_1h_2\in H$ whenever $h_1$ and $h_2$ belong to $H.$