Jacobian of a diffeomorphism

Question

Let $U,V\subseteq \mathbb{R}^{n}$ be open. Let $\alpha:U \to V$ be a smooth homeomorphism. Furthermore, assume that $\mathcal{J}_{\alpha}(\mathbf{x})$ (the Jacobian matrix) has rank $n$ for all $\mathbf{x} \in U$.

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Is it true that $\det\left( \mathcal{J}_{\alpha}(\mathbf{x}) \right) \neq 0$ for all $\mathbf{x} \in U$?

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If so, how do I prove this. If not, why not?

Answer 1

Hint:

if an $n\times n$ matrix has rank $n$ that it is invertible (see here) so its determinant in not null.