Given a $2\times2$ matrix $A$ with entries in $\mathbb Z$, it acts on \begin{equation} \mathbb T^2=\mathbb R^2/\mathbb Z^2 \end{equation} and then leads to a map \begin{equation} A_*\colon H_2(\mathbb T^2)\to H_2(\mathbb T^2) \end{equation}
I want to know is this map just the scalar multiplication by $\det A$?
Can someone suggest a neat proof?
Thank you in advance.