Given $A:\mathbb R^n \rightarrow \mathbb R^{n\times n}$, $x\mapsto A(x)$ invertible for all $x$. In particular, it is known that $$A(x) = \frac{\partial}{\partial x} f(x)$$ with $f:\mathbb R^n \rightarrow \mathbb R^n$, continuously differentiable and $f(0) = 0$.
Show that$$T(x) := \int_0^1 A(\sigma x) d\sigma $$ is invertible for all $x \in \mathbb R^n$.