grandes-ecoles 2016 QIII.C

grandes-ecoles · France · centrale-maths1__psi Matrices Matrix Group and Subgroup Structure

We are given a matrix $M$ of $\mathrm{GL}_n(\mathbb{R})$ whose coefficients are all natural integers and such that the set formed by all coefficients of all successive powers of $M$ is finite.
Prove that $M^{-1}$ has coefficients in $\mathbb{N}$ and deduce that $M$ is a permutation matrix. What can be said of the converse?

We are given a matrix $M$ of $\mathrm{GL}_n(\mathbb{R})$ whose coefficients are all natural integers and such that the set formed by all coefficients of all successive powers of $M$ is finite.

Prove that $M^{-1}$ has coefficients in $\mathbb{N}$ and deduce that $M$ is a permutation matrix. What can be said of the converse?

Paper Questions

QI.A.1 QI.A.2 QI.A.3 QI.A.4 QI.B.1 QI.B.2 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.B.4 QIII.C QIV.A.1 QIV.A.2 QIV.A.3 QIV.A.4 QIV.A.5 QIV.A.6 QIV.A.7 QIV.B.1 QIV.B.2 QIV.B.3 QIV.B.4 QIV.B.5 QIV.B.6