grandes-ecoles 2013 QII.A.2

grandes-ecoles · France · centrale-maths2__psi Matrices Matrix Entry and Coefficient Identities

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 real number $\theta_n$ such that $$\frac{1}{\beta_n}\left(I_2 + \frac{1}{n}A\right) = \begin{pmatrix} \cos\theta_n & -\sin\theta_n \\ \sin\theta_n & \cos\theta_n \end{pmatrix}$$

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 real number $\theta_n$ such that
$$\frac{1}{\beta_n}\left(I_2 + \frac{1}{n}A\right) = \begin{pmatrix} \cos\theta_n & -\sin\theta_n \\ \sin\theta_n & \cos\theta_n \end{pmatrix}$$

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