grandes-ecoles 2016 QII.A

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

Let $A$ denote a positive matrix in $\mathcal{M}_n(\mathbb{R})$.
Show that if there exists in $A$ a path from $i$ to $j$, with $i \neq j$, then there exists an elementary path from $i$ to $j$ of length $\ell \leqslant n-1$. Similarly, show that if there exists in $A$ a circuit passing through $i$, then there exists an elementary circuit passing through $i$ of length $\ell \leqslant n$.

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

Show that if there exists in $A$ a path from $i$ to $j$, with $i \neq j$, then there exists an elementary path from $i$ to $j$ of length $\ell \leqslant n-1$. Similarly, show that if there exists in $A$ a circuit passing through $i$, then there exists 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