Prove that Aut($\mathbb Z \times \mathbb Z$) $\cong$ $\text{GL}_2(\mathbb Z)$.
This is a HW problem for an Algebra course, hints/suggestions welcome.
I didn't find this problem on math.SE, however I did find a partial solution in the course notes of a different university, which gave me the general form of the argument.
My attempt is below. There are a couple details I think I don't understand fully, and to me the statements sound a bit fuzzy - I've indicated these with $\star \dots \star$.
$\DeclareMathOperator{\aut}{Aut}$ Recall that automorphisms are completely determined by the images of generators. Write an element $(a,b) \in \mathbb Z\times \mathbb Z$ as a column vector $\left( \begin{smallmatrix} a \\ b \end{smallmatrix}\right)$ and let $\varphi_M$ be an automorphism of $\mathbb Z\times \mathbb Z$ given by $\varphi_M \left( \left( \begin{smallmatrix} 1\\ 0 \end{smallmatrix} \right) \right)= \left( \begin{smallmatrix} a\\ b \end{smallmatrix} \right)$ and $\varphi_M \left( \left( \begin{smallmatrix} 0\\ 1 \end{smallmatrix} \right) \right)= \left( \begin{smallmatrix} c\\ d \end{smallmatrix} \right)$ for some integers $a,b,c,d$. $\star$ Note that $\varphi_M$ therefore corresponds to left multiplication by the matrix $M := \left( \begin{smallmatrix} a & b \\ c & d \end{smallmatrix} \right)$. $\star$
Since $\varphi_M$ is an automorphism, it has an inverse $\varphi_{M^{-1}}$ associated to a matrix $M^{-1}$ such that $MM^{-1}=I$. It follows that $\det M \det M^{-1} = (ad-bc)\det M^{-1} =1$, and therefore $\det M = \pm 1$, which implies that $M\in \operatorname{GL}_2 (\mathbb Z)$. Conversely, a matrix $N\in \operatorname{GL}_2(\mathbb Z)$ is invertible, so it corresponds to an automorphism $\varphi_N \in \aut(\mathbb Z\times \mathbb Z)$. Thus we have a bijection $\aut(\mathbb Z\times \mathbb Z)\to \operatorname{GL}_2(\mathbb Z)$, $\star$ so to get an isomorphism it suffices to observe that composition of automorphisms corresponds to matrix multiplication. $\star$ Hence $\aut(\mathbb Z\times \mathbb Z)\cong \operatorname{GL}_2(\mathbb Z).$
Is this argument correct, and how can I fill in the details of the $\star$ sentences with a more precise statement?
Edit: I think it's also worth asking if there is a more streamlined approach to answering this question?