grandes-ecoles 2014 QVE

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 $$\forall A \in \mathcal{M}_d(\mathbb{R}), \quad A \text{ invertible} \Rightarrow f_\xi(A) = \left(\xi(A_{i,j})\right)_{1\leqslant i,j\leqslant d} \text{ invertible} \tag{V.1}$$
The purpose of this question is to show $\xi(0) = 0$.
1) Show that if $\xi(0) \neq 0$, then there exists $\alpha > 0$ such that $\xi(0)\xi(2) = \xi(1)\xi(\alpha)$.
2) Conclude.

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

The purpose of this question is to show $\xi(0) = 0$.

1) Show that if $\xi(0) \neq 0$, then there exists $\alpha > 0$ such that $\xi(0)\xi(2) = \xi(1)\xi(\alpha)$.

2) Conclude.

Paper Questions

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