grandes-ecoles 2016 QV.A.1

grandes-ecoles · France · centrale-maths1__mp 3x3 Matrices Matrix Algebraic Properties and Abstract Reasoning

Let $A = (a_{i,j})$ in $\mathcal{M}_n(\mathbb{R})$, with $A \geqslant 0$. We say that $A$ is irreducible if, for all $i$ and $j$ in $\llbracket 1, n \rrbracket$, there exists $m \geqslant 0$ (depending a priori on $i$ and $j$) such that $a_{i,j}^{(m)} > 0$.
Express the irreducibility of $A$ in terms of paths in $A$.

Let $A = (a_{i,j})$ in $\mathcal{M}_n(\mathbb{R})$, with $A \geqslant 0$. We say that $A$ is irreducible if, for all $i$ and $j$ in $\llbracket 1, n \rrbracket$, there exists $m \geqslant 0$ (depending a priori on $i$ and $j$) such that $a_{i,j}^{(m)} > 0$.

Express the irreducibility of $A$ in terms of paths in $A$.

Paper Questions

QI.A.1 QI.A.2 QI.B.1 QI.B.2 QII.A QII.B QII.C QIII.A QIII.B.1 QIII.B.2 QIII.B.3 QIII.B.4 QIII.B.5 QIII.C.1 QIII.C.2 QIII.C.3 QIII.D.1 QIII.D.2 QIV.A.1 QIV.A.2 QIV.A.3 QIV.A.4 QIV.B.1 QIV.B.2 QIV.B.3 QIV.C QV.A.1 QV.A.2 QV.A.3 QV.A.4 QV.A.5 QV.B.1 QV.B.2 QV.C.1 QV.C.2 QVI.A QVI.B.1 QVI.B.2 QVI.B.3 QVI.C.1 QVI.C.2 QVI.D