Prove that $GL_2(\mathbb{Z})$ is not a subgroup of $GL_2(\mathbb{R})$

Question

$GL_2(\mathbb{R})=\left\lbrace \begin{bmatrix}a&b\\c&d\end{bmatrix} \,\middle|\, a,b,c,d\in\mathbb{R}\textrm{ and } ad-bc\neq 0\right\rbrace$

Claim: The subset $GL_2(\mathbb{Z})=\left\lbrace \begin{bmatrix}a&b\\c&d\end{bmatrix} \,\middle|\, a,b,c,d\in\mathbb{Z}\textrm{ and } ad-bc\neq 0\right\rbrace$ is not a subgroup.

I'm not really sure how to show this is not a subgroup. I believe it may have something to do with the inverse perhaps having a result in $\mathbb{R}$. Because I believe this group satisfies the closure, associativity and identity axioms. But, i'm not entirely sure.

Answer 1

For instance, $\left[\begin{smallmatrix}1&0\\0&2\end{smallmatrix}\right]\in GL_2(\mathbb Z)$, but $\left[\begin{smallmatrix}1&0\\0&2\end{smallmatrix}\right]^{-1}\notin GL_2(\mathbb Z)$.