grandes-ecoles 2016 QIII.A

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

Let $A$ be a primitive matrix in $\mathcal{M}_n(\mathbb{R})$.
Show that for all $i \neq j$ there exists in $A$ an elementary path from $i$ to $j$ of length $\ell \leqslant n-1$, and that for all $i$ there exists in $A$ an elementary circuit passing through $i$ of length $\ell \leqslant n$.

Let $A$ be a primitive matrix in $\mathcal{M}_n(\mathbb{R})$.

Show that for all $i \neq j$ there exists in $A$ an elementary path from $i$ to $j$ of length $\ell \leqslant n-1$, and that for all $i$ there exists in $A$ an elementary circuit passing through $i$ of length $\ell \leqslant n$.

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