Degree of a map $T^2\to T^2$ induced by an $2\times 2$ integer matrix

There's a natural map $\wedge^2(H^1(T^2, \mathbb{Z})) \to H^2(T^2, \mathbb{Z})$ given by the cup product. Show that it's an isomorphism. Then the desired result follows from the fact that $\det(A)$ is the scalar by which $A$ acts on the top exterior power, together with the universal coefficient theorem.