grandes-ecoles 2013 QII.A.1

grandes-ecoles · France · centrale-maths2__psi Matrices Matrix Group and Subgroup Structure

Let $n \in \mathbb{N}^*$. Let $A \in M_2(\mathbb{R})$ be an antisymmetric matrix. We set $A = \begin{pmatrix} 0 & -\alpha \\ \alpha & 0 \end{pmatrix}$ where $\alpha \in \mathbb{R}$.
Determine a number $\beta_n \in \mathbb{R}_+^*$ such that $$\frac{1}{\beta_n}\left(I_2 + \frac{1}{n}A\right) \in SO_2(\mathbb{R})$$

Let $n \in \mathbb{N}^*$. Let $A \in M_2(\mathbb{R})$ be an antisymmetric matrix. We set $A = \begin{pmatrix} 0 & -\alpha \\ \alpha & 0 \end{pmatrix}$ where $\alpha \in \mathbb{R}$.

Determine a number $\beta_n \in \mathbb{R}_+^*$ such that
$$\frac{1}{\beta_n}\left(I_2 + \frac{1}{n}A\right) \in SO_2(\mathbb{R})$$

Paper Questions

QI.A QI.B QII.A.1 QII.A.2 QII.A.3 QII.B.1 QII.B.2 QII.B.3 QII.B.4 QIII.A.1 QIII.A.2 QIII.A.3 QIII.B.1 QIII.B.2 QIII.B.3 QIII.C.1 QIII.C.2 QIV.A.1 QIV.A.2 QIV.B QIV.C QIV.D QIV.E QIV.F QV.A.1 QV.A.2 QV.A.3 QV.A.4 QV.B.1 QV.B.2 QV.B.3 QV.B.4