grandes-ecoles 2014 QVD

grandes-ecoles · France · centrale-maths1__mp Composite & Inverse Functions Existence or Properties of Functions and Inverses (Proof-Based)

We are given a continuous function $\xi : \mathbb{R} \rightarrow \mathbb{R}$ satisfying condition (V.1) (with $d \geqslant 2$), where $$\forall A \in \mathcal{M}_d(\mathbb{R}), \quad A \text{ invertible} \Rightarrow f_\xi(A) = \left(\xi(A_{i,j})\right)_{1\leqslant i,j\leqslant d} \text{ invertible} \tag{V.1}$$
Show that the function $\xi$ does not vanish on $\mathbb{R}^*$.

We are given a continuous function $\xi : \mathbb{R} \rightarrow \mathbb{R}$ satisfying condition (V.1) (with $d \geqslant 2$), where
$$\forall A \in \mathcal{M}_d(\mathbb{R}), \quad A \text{ invertible} \Rightarrow f_\xi(A) = \left(\xi(A_{i,j})\right)_{1\leqslant i,j\leqslant d} \text{ invertible} \tag{V.1}$$

Show that the function $\xi$ does not vanish on $\mathbb{R}^*$.

Paper Questions

QIA QIB QIC QID QIE QIIA QIIB QIIC QIIIA1 QIIIA2 QIIIA3 QIIIB1 QIIIB2 QIIIB3 QIIIB4 QIVA QIVB QIVC QVA QVB QVC QVD QVE QVF QVG QVH QVI QVJ