grandes-ecoles 2016 QII.B

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

Let $A \geqslant 0$ in $\mathcal{M}_n(\mathbb{R})$. Let $i, j$ be in $\llbracket 1, n \rrbracket$. Let $m \geqslant 1$. Show the equivalence of the propositions:

there exists in $A$ a path with origin $i$, endpoint $j$, of length $m$;
the entry with indices $i, j$ of $A^m$ (denoted $a_{i,j}^{(m)}$) is strictly positive.

You may proceed by induction on the integer $m \geqslant 1$.

Let $A \geqslant 0$ in $\mathcal{M}_n(\mathbb{R})$. Let $i, j$ be in $\llbracket 1, n \rrbracket$. Let $m \geqslant 1$. Show the equivalence of the propositions:
\begin{itemize}
  \item there exists in $A$ a path with origin $i$, endpoint $j$, of length $m$;
  \item the entry with indices $i, j$ of $A^m$ (denoted $a_{i,j}^{(m)}$) is strictly positive.
\end{itemize}
You may proceed by induction on the integer $m \geqslant 1$.

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