Confused about what to show a matrix is an element of GL(2,$\mathbb{R}$).

Question

I am kind of confused about what I am supposed to show in the following problem:

Show that for any $\theta \in \mathbb{R}$, we have $$ \begin{pmatrix} cos \hspace1mm\theta& -sin \hspace1mm\theta \\ sin \hspace1mm \theta & cos \hspace1mm\theta \\ \end{pmatrix} \in GL(2,\mathbb{R})$$

Am I supposed to show that matrices of this form satisfy closure, associativity, existence of an identity and inverse? That is for matrices of this form, but with different $\theta$'s? I am not to sure what exactly it is asking.

Answer 1

Since $GL(2,\mathbb R) = \{ A\in M_2(\mathbb R) : A$ is invertible$\}$, all you have to prove is that these matrices have determinant other than $0$.