grandes-ecoles 2013 QIV.A.2

grandes-ecoles · France · centrale-maths2__psi Matrices Linear Transformation and Endomorphism Properties

Let $A \in M_p(\mathbb{C})$ and $k \in \mathbb{N}^*$ such that $A^k = 0$ and $A^{k-1} \neq 0$ (we say that $A$ is nilpotent of order $k$).
Deduce that $k \leqslant p$.

Let $A \in M_p(\mathbb{C})$ and $k \in \mathbb{N}^*$ such that $A^k = 0$ and $A^{k-1} \neq 0$ (we say that $A$ is nilpotent of order $k$).

Deduce that $k \leqslant p$.

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