Apologies in advance for the poor formatting.
Let G be the following set of matrices
\begin{bmatrix}\cos(x)&\sin(x)\\-\sin(x)&\cos(x)\end{bmatrix}
union
\begin{bmatrix}\cos(x)&\sin(x)\\\sin(x)&-\cos(x)\end{bmatrix}
where x is an element of the real numbers.
Show that G is a group under matrix multiplication.
Now I know I need to check the group axioms and, barring the closure one, the rest are pretty easy. There was just a bit of confusion with the closure axiom though. Since I have a union here, it won't suffice to just check whether the product of the two matrices are in G right? I'd also have to check the product of the first one by itself, the second one by itself and the second one multiplied by the first one? Or am I barking up the wrong tree here?
Any help here would be much appreciated.
Best Answer
You're exactly right. To check closure properly you have to check all permutations of multiplying the two together, and ensure that out pops something in the form of one of the two.