grandes-ecoles 2014 QIIIA3

grandes-ecoles · France · centrale-maths1__mp Matrices Matrix Algebra and Product Properties

For every $A \in \mathcal{M}_d(\mathbb{R})$, we define $$\cos(A) = \sum_{n=0}^{+\infty} (-1)^n \frac{A^{2n}}{(2n)!} \quad \text{and} \quad \sin(A) = \sum_{n=0}^{+\infty} (-1)^n \frac{A^{2n+1}}{(2n+1)!}$$ Show $$\forall A \in \mathcal{M}_d(\mathbb{R}), \quad \cos(A)^2 + \sin(A)^2 = I_d$$

For every $A \in \mathcal{M}_d(\mathbb{R})$, we define
$$\cos(A) = \sum_{n=0}^{+\infty} (-1)^n \frac{A^{2n}}{(2n)!} \quad \text{and} \quad \sin(A) = \sum_{n=0}^{+\infty} (-1)^n \frac{A^{2n+1}}{(2n+1)!}$$
Show
$$\forall A \in \mathcal{M}_d(\mathbb{R}), \quad \cos(A)^2 + \sin(A)^2 = I_d$$

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