Square matrix/Z/Bijection with determinant/Exercise

Let ${}M$ be an ${}n\times n$-matrix with entries in ${}\mathbb {Z}$, and let

$\varphi _{M}\colon \mathbb {Z} ^{n}\longrightarrow \mathbb {Z} ^{n}$

denote the corresponding group homomorphism. Show that ${}\varphi _{M}$ is bijective if and only if the determinant of ${}M$ equals ${}1$ or equals ${}-1$.

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