grandes-ecoles

Papers (176)

2025

centrale-maths1__official 40 centrale-maths2__official 36 mines-ponts-maths1__mp 17 mines-ponts-maths1__pc 21 mines-ponts-maths1__psi 21 mines-ponts-maths2__mp 28 mines-ponts-maths2__pc 23 mines-ponts-maths2__psi 25 polytechnique-maths-a__mp 35 polytechnique-maths__fui 9 polytechnique-maths__pc 27 x-ens-maths-a__fui 10 x-ens-maths-a__mp 18 x-ens-maths-b__mp 6 x-ens-maths-c__mp 6 x-ens-maths-d__mp 31 x-ens-maths__pc 27 x-ens-maths__psi 30

2024

centrale-maths1__official 21 centrale-maths2__official 28 geipi-polytech__maths 9 mines-ponts-maths1__mp 23 mines-ponts-maths1__psi 9 mines-ponts-maths2__mp 14 mines-ponts-maths2__pc 19 mines-ponts-maths2__psi 20 polytechnique-maths-a__mp 42 polytechnique-maths-b__mp 27 x-ens-maths-a__mp 43 x-ens-maths-b__mp 29 x-ens-maths-c__mp 22 x-ens-maths-d__mp 41 x-ens-maths__pc 20 x-ens-maths__psi 23

2023

centrale-maths1__official 37 centrale-maths2__official 32 e3a-polytech-maths__mp 4 mines-ponts-maths1__mp 14 mines-ponts-maths1__pc 21 mines-ponts-maths1__psi 21 mines-ponts-maths2__mp 21 mines-ponts-maths2__pc 13 mines-ponts-maths2__psi 22 polytechnique-maths__fui 3 x-ens-maths-a__mp 24 x-ens-maths-b__mp 10 x-ens-maths-c__mp 10 x-ens-maths-d__mp 10 x-ens-maths__pc 22

2022

centrale-maths1__mp 22 centrale-maths1__pc 33 centrale-maths1__psi 42 centrale-maths2__mp 26 centrale-maths2__pc 37 centrale-maths2__psi 40 mines-ponts-maths1__mp 26 mines-ponts-maths1__pc 20 mines-ponts-maths1__psi 23 mines-ponts-maths2__mp 22 mines-ponts-maths2__pc 9 mines-ponts-maths2__psi 18 x-ens-maths-a__mp 8 x-ens-maths-b__mp 19 x-ens-maths-c__mp 17 x-ens-maths-d__mp 47 x-ens-maths1__mp 13 x-ens-maths2__mp 26 x-ens-maths__pc 7 x-ens-maths__pc_cpge 14 x-ens-maths__psi 22 x-ens-maths__psi_cpge 26

2021

centrale-maths1__mp 34 centrale-maths1__pc 36 centrale-maths1__psi 28 centrale-maths2__mp 21 centrale-maths2__pc 38 centrale-maths2__psi 28 x-ens-maths2__mp 35 x-ens-maths__pc 29

2020

centrale-maths1__mp 42 centrale-maths1__pc 36 centrale-maths1__psi 38 centrale-maths2__mp 2 centrale-maths2__pc 35 centrale-maths2__psi 39 mines-ponts-maths1__mp_cpge 22 mines-ponts-maths2__mp_cpge 19 x-ens-maths-a__mp_cpge 10 x-ens-maths-b__mp_cpge 19 x-ens-maths-c__mp 10 x-ens-maths-d__mp 13 x-ens-maths1__mp 13 x-ens-maths2__mp 20 x-ens-maths__pc 6

2019

centrale-maths1__mp 37 centrale-maths1__pc 40 centrale-maths1__psi 38 centrale-maths2__mp 37 centrale-maths2__pc 39 centrale-maths2__psi 46 x-ens-maths1__mp 24 x-ens-maths__pc 18 x-ens-maths__psi 9

2018

centrale-maths1__mp 21 centrale-maths1__pc 31 centrale-maths1__psi 39 centrale-maths2__mp 23 centrale-maths2__pc 35 centrale-maths2__psi 30 x-ens-maths1__mp 18 x-ens-maths2__mp 13 x-ens-maths__pc 17 x-ens-maths__psi 20

2017

centrale-maths1__mp 45 centrale-maths1__pc 22 centrale-maths1__psi 17 centrale-maths2__mp 30 centrale-maths2__pc 28 centrale-maths2__psi 44 x-ens-maths1__mp 24 x-ens-maths2__mp 7 x-ens-maths__pc 17 x-ens-maths__psi 19

2016

centrale-maths1__mp 41 centrale-maths1__pc 31 centrale-maths1__psi 33 centrale-maths2__mp 25 centrale-maths2__pc 42 centrale-maths2__psi 17 x-ens-maths1__mp 10 x-ens-maths2__mp 32 x-ens-maths__pc 1 x-ens-maths__psi 20

2015

centrale-maths1__mp 18 centrale-maths1__pc 11 centrale-maths1__psi 42 centrale-maths2__mp 44 centrale-maths2__pc 1 centrale-maths2__psi 14 x-ens-maths1__mp 16 x-ens-maths2__mp 19 x-ens-maths__pc 30 x-ens-maths__psi 20

2014

centrale-maths1__mp 28 centrale-maths1__pc 26 centrale-maths1__psi 36 centrale-maths2__mp 24 centrale-maths2__pc 23 centrale-maths2__psi 29 x-ens-maths2__mp 13

2013

centrale-maths1__mp 3 centrale-maths1__pc 45 centrale-maths1__psi 20 centrale-maths2__mp 32 centrale-maths2__pc 50 centrale-maths2__psi 32 x-ens-maths1__mp 14 x-ens-maths2__mp 10 x-ens-maths__pc 22 x-ens-maths__psi 9

2012

centrale-maths1__pc 23 centrale-maths1__psi 20 centrale-maths2__mp 27 centrale-maths2__psi 20

2011

centrale-maths1__mp 27 centrale-maths1__pc 15 centrale-maths1__psi 21 centrale-maths2__mp 29 centrale-maths2__pc 8 centrale-maths2__psi 28

2010

centrale-maths1__mp 7 centrale-maths1__pc 23 centrale-maths1__psi 9 centrale-maths2__mp 10 centrale-maths2__pc 36 centrale-maths2__psi 27

2021 centrale-maths1__psi

28 maths questions

Q1 Probability Definitions Direct Proof of a Stated Identity or Equality View

Justify that, for every natural integer $k$, $p _ { 1 } ^ { ( k ) } + \cdots + p _ { n } ^ { ( k ) } = 1$.

Q2 Matrices Matrix Power Computation and Application View

Show that, for every natural integer $k$, $P ^ { ( k + 1 ) } = P ^ { ( k ) } T$.

Q3 Matrices Closed-form expression derivation View

Deduce from this, for every natural integer $k$, an expression for $P ^ { ( k ) }$ as a function of $T$, $k$ and $P ^ { ( 0 ) }$.

Q4 Matrices Linear Transformation and Endomorphism Properties View

Suppose that the sequence of vectors $\left( P ^ { ( k ) } \right) _ { k \in \mathbb { N } }$ converges to a vector $P = \left( p _ { 1 } , \ldots , p _ { n } \right)$. Show that $P T = P$, that for all $i \in \llbracket 1 , n \rrbracket$, $p _ { i } \geqslant 0$ and that $p _ { 1 } + \cdots + p _ { n } = 1$.

Q5 Matrices Structured Matrix Characterization View

We consider the directed graph $G = ( S , A )$ where $$\left\{ \begin{array} { l } S = \{ 1,2,3,4 \} \\ A = \{ ( 1,2 ) , ( 2,1 ) , ( 1,3 ) , ( 3,1 ) , ( 1,4 ) , ( 4,1 ) , ( 2,3 ) , ( 3,2 ) , ( 2,4 ) , ( 4,2 ) , ( 3,4 ) , ( 4,3 ) \} \end{array} \right.$$ We assume that, when the point is on one of the vertices of the graph, it has the same probability of going to each of the three other vertices of the graph. We set $$J _ { 4 } = \left( \begin{array} { l l l l } 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 \end{array} \right)$$ Express the transition matrix $T$ as a function of $J _ { 4 }$ and $I _ { 4 }$.

Q6 Matrices Matrix Decomposition and Factorization View

We consider the directed graph $G = ( S , A )$ where $$\left\{ \begin{array} { l } S = \{ 1,2,3,4 \} \\ A = \{ ( 1,2 ) , ( 2,1 ) , ( 1,3 ) , ( 3,1 ) , ( 1,4 ) , ( 4,1 ) , ( 2,3 ) , ( 3,2 ) , ( 2,4 ) , ( 4,2 ) , ( 3,4 ) , ( 4,3 ) \} \end{array} \right.$$ We assume that, when the point is on one of the vertices of the graph, it has the same probability of going to each of the three other vertices of the graph. We set $$J _ { 4 } = \left( \begin{array} { l l l l } 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 \end{array} \right)$$ Prove that there exists a matrix $Q \in \mathcal { O } _ { 4 } ( \mathbb { R } )$ such that $$T = \frac { 1 } { 3 } Q \left( \begin{array} { c c c c } - 1 & 0 & 0 & 0 \\ 0 & - 1 & 0 & 0 \\ 0 & 0 & - 1 & 0 \\ 0 & 0 & 0 & 3 \end{array} \right) Q ^ { \top }$$

Q7 Matrices Matrix Power Computation and Application View

We consider the directed graph $G = ( S , A )$ where $$\left\{ \begin{array} { l } S = \{ 1,2,3,4 \} \\ A = \{ ( 1,2 ) , ( 2,1 ) , ( 1,3 ) , ( 3,1 ) , ( 1,4 ) , ( 4,1 ) , ( 2,3 ) , ( 3,2 ) , ( 2,4 ) , ( 4,2 ) , ( 3,4 ) , ( 4,3 ) \} \end{array} \right.$$ We assume that, when the point is on one of the vertices of the graph, it has the same probability of going to each of the three other vertices of the graph. Show that the sequence of matrices $\left( T ^ { k } \right) _ { k \in \mathbb { N } }$ converges and identify geometrically the endomorphism canonically associated with the limit matrix.

Q8 Matrices Matrix Power Computation and Application View

We consider the directed graph $G = ( S , A )$ where $$\left\{ \begin{array} { l } S = \{ 1,2,3,4 \} \\ A = \{ ( 1,2 ) , ( 2,1 ) , ( 1,3 ) , ( 3,1 ) , ( 1,4 ) , ( 4,1 ) , ( 2,3 ) , ( 3,2 ) , ( 2,4 ) , ( 4,2 ) , ( 3,4 ) , ( 4,3 ) \} \end{array} \right.$$ We assume that, when the point is on one of the vertices of the graph, it has the same probability of going to each of the three other vertices of the graph. Show that, for any row vector $P ^ { ( 0 ) } = \left( p _ { 1 } ^ { ( 0 ) } , p _ { 2 } ^ { ( 0 ) } , p _ { 3 } ^ { ( 0 ) } , p _ { 4 } ^ { ( 0 ) } \right)$, where for $1 \leqslant i \leqslant 4$, $p _ { i } ^ { ( 0 ) }$ is the probability that the point is initially on vertex $i$, the sequence $\left( P ^ { ( k ) } \right) _ { k \in \mathbb { N } }$ converges to the row vector $( 1 / 4,1 / 4,1 / 4,1 / 4 )$.

Q9 Matrices Matrix Algebra and Product Properties View

We consider the graph $G$ represented in Figure 2. We recall that, when the point is on one of the vertices of the graph, it has the same probability of going to each of the vertices to which it is connected. We assume that initially, the point is on vertex 1, so that $P ^ { ( 0 ) } = ( 1,0,0,0,0,0,0,0 )$. We denote $S _ { 1 } = \{ 1,3,6,8 \}$ and $S _ { 2 } = \{ 2,4,5,7 \}$.
Give the transition matrix $T$ of this graph and calculate $$( 1,1,1,1,1,1,1,1 ) T .$$

Q10 Matrices Proof of Distributional Properties or Symmetry View

We consider the graph $G$ represented in Figure 2. We denote $S _ { 1 } = \{ 1,3,6,8 \}$ and $S _ { 2 } = \{ 2,4,5,7 \}$.
Show that, if the point is on a vertex of part $S _ { 1 }$ at a given step, it will be on a vertex of part $S _ { 2 }$ at the next step and that, if it is on a vertex of $S _ { 2 }$ at a given step, it will be on a vertex of $S _ { 1 }$ at the next step.

Q11 Matrices Markov Chain and Transition Matrix Analysis View

We consider the graph $G$ represented in Figure 2. We denote $S _ { 1 } = \{ 1,3,6,8 \}$ and $S _ { 2 } = \{ 2,4,5,7 \}$. We assume that initially, the point is on vertex 1, so that $P ^ { ( 0 ) } = ( 1,0,0,0,0,0,0,0 )$.
Is the sequence of vectors $\left( P ^ { ( k ) } \right) _ { k \in \mathbb { N } }$ convergent?

Q12 Matrices Linear System and Inverse Existence View

Let $M \in M _ { n } ( \mathbb { R } )$, all of whose coefficients are non-negative. Show that $M$ is a stochastic matrix if and only if $$M \left( \begin{array} { c } 1 \\ \vdots \\ 1 \end{array} \right) = \left( \begin{array} { c } 1 \\ \vdots \\ 1 \end{array} \right) .$$

Q13 Matrices Linear System and Inverse Existence View

Show that the transition matrix of a graph (defined in part I) is a stochastic matrix and that, for every natural integer $k$, the vector $P ^ { ( k ) }$, also defined in part I, is a probability distribution.

Q14 Matrices Linear Transformation and Endomorphism Properties View

Let $M \in \mathcal { M } _ { n } ( \mathbb { R } )$ and $N \in \mathcal { M } _ { n } ( \mathbb { R } )$ be two stochastic matrices, $X \in \mathbb { R } ^ { n }$ a probability distribution and $\alpha \in [ 0,1 ]$. Show that $X M$ is a probability distribution.

Q15 Matrices Matrix Algebra and Product Properties View

Q16 Matrices Matrix Algebra and Product Properties View

Q17 Matrices Eigenvalue constraints from matrix properties View

Let $M = \left( m _ { i , j } \right)$ be a stochastic matrix of $\mathcal { M } _ { n } ( \mathbb { R } )$ and $\lambda$ an eigenvalue (real or complex) of $M$. We denote by $\left( u _ { 1 } , \ldots , u _ { n } \right)$ the components (real or complex), in the canonical basis, of an eigenvector $u$ associated with $\lambda$.
Let $h \in \{ 1 , \ldots , n \}$ such that $\left| u _ { h } \right| = \max _ { 1 \leqslant i \leqslant n } \left| u _ { i } \right|$. Show that $\left| \lambda - m _ { h , h } \right| \leqslant 1 - m _ { h , h }$. Deduce that $| \lambda | \leqslant 1$.

Q18 Matrices Eigenvalue constraints from matrix properties View

Let $M = \left( m _ { i , j } \right)$ be a stochastic matrix of $\mathcal { M } _ { n } ( \mathbb { R } )$ and $\lambda$ an eigenvalue (real or complex) of $M$. We denote by $\left( u _ { 1 } , \ldots , u _ { n } \right)$ the components (real or complex), in the canonical basis, of an eigenvector $u$ associated with $\lambda$.
Let $\delta = \min _ { 1 \leqslant i \leqslant n } m _ { i , i }$. Show that $| \lambda - \delta | \leqslant 1 - \delta$. Give a geometric interpretation of this result and show that, if all diagonal terms of $M$ are strictly positive, then 1 is the only eigenvalue of $M$ with modulus 1.

Q19 Matrices Linear Transformation and Endomorphism Properties View

We now assume that all coefficients $m _ { i , j } ( 1 \leqslant i , j \leqslant n )$ of the stochastic matrix $M$ are strictly positive.
Prove that $\operatorname { dim } \left( \operatorname { ker } \left( M - I _ { n } \right) \right) = 1$.
If $\left( u _ { 1 } , \ldots , u _ { n } \right)$ denotes the components (real) in the canonical basis of a vector of $\operatorname { ker } \left( M - I _ { n } \right)$, one may use $\min _ { 1 \leqslant i \leqslant n } u _ { i }$.

Q20 Matrices Linear Transformation and Endomorphism Properties View

We now assume that all coefficients $m _ { i , j } ( 1 \leqslant i , j \leqslant n )$ of the stochastic matrix $M$ are strictly positive.
Deduce that there exists at most one probability distribution $X$ invariant by $M$, that is, satisfying $X M = X$.

Q21 Matrices Matrix Power Computation and Application View

We now assume that all coefficients $m _ { i , j } ( 1 \leqslant i , j \leqslant n )$ of the stochastic matrix $M$ are strictly positive. We set $\varepsilon = \min _ { 1 \leqslant i , j \leqslant n } m _ { i , j }$. We are interested in the sequence $\left( M ^ { k } \right) _ { k \in \mathbb { N } }$ of powers of $M$. We denote by $m _ { i , j } ^ { ( k ) }$ the coefficient of the matrix $M ^ { k }$ located in row $i$ and column $j$.
For all $j \in \llbracket 1 , n \rrbracket$, we set $$\left\{ \begin{array} { l } \alpha _ { j } ^ { ( k ) } = \min _ { 1 \leqslant i \leqslant n } m _ { i , j } ^ { ( k ) } , \\ \beta _ { j } ^ { ( k ) } = \max _ { 1 \leqslant i \leqslant n } m _ { i , j } ^ { ( k ) } . \end{array} \right.$$
In the following four questions, $j$ is a fixed integer in $\llbracket 1 , n \rrbracket$ and $k$ is fixed in $\mathbb { N }$. Prove the inequalities $\alpha _ { j } ^ { ( k ) } \leqslant \alpha _ { j } ^ { ( k + 1 ) } \leqslant \beta _ { j } ^ { ( k + 1 ) } \leqslant \beta _ { j } ^ { ( k ) }$.

Q22 Matrices Matrix Entry and Coefficient Identities View

We now assume that all coefficients $m _ { i , j } ( 1 \leqslant i , j \leqslant n )$ of the stochastic matrix $M$ are strictly positive. We set $\varepsilon = \min _ { 1 \leqslant i , j \leqslant n } m _ { i , j }$. We are interested in the sequence $\left( M ^ { k } \right) _ { k \in \mathbb { N } }$ of powers of $M$. We denote by $m _ { i , j } ^ { ( k ) }$ the coefficient of the matrix $M ^ { k }$ located in row $i$ and column $j$.
For all $j \in \llbracket 1 , n \rrbracket$, we set $$\left\{ \begin{array} { l } \alpha _ { j } ^ { ( k ) } = \min _ { 1 \leqslant i \leqslant n } m _ { i , j } ^ { ( k ) } , \\ \beta _ { j } ^ { ( k ) } = \max _ { 1 \leqslant i \leqslant n } m _ { i , j } ^ { ( k ) } . \end{array} \right.$$
In the following four questions, $j$ is a fixed integer in $\llbracket 1 , n \rrbracket$ and $k$ is fixed in $\mathbb { N }$. Prove that there exists a pair $\left( i _ { 0 } , j _ { 0 } \right) \in \llbracket 1 , n \rrbracket ^ { 2 }$ such that $$\alpha _ { j } ^ { ( k + 1 ) } - \alpha _ { j } ^ { ( k ) } \geqslant m _ { i _ { 0 } , j _ { 0 } } \left( \beta _ { j } ^ { ( k ) } - \alpha _ { j } ^ { ( k ) } \right) .$$

Q23 Matrices Matrix Entry and Coefficient Identities View

We now assume that all coefficients $m _ { i , j } ( 1 \leqslant i , j \leqslant n )$ of the stochastic matrix $M$ are strictly positive. We set $\varepsilon = \min _ { 1 \leqslant i , j \leqslant n } m _ { i , j }$. We are interested in the sequence $\left( M ^ { k } \right) _ { k \in \mathbb { N } }$ of powers of $M$. We denote by $m _ { i , j } ^ { ( k ) }$ the coefficient of the matrix $M ^ { k }$ located in row $i$ and column $j$.
For all $j \in \llbracket 1 , n \rrbracket$, we set $$\left\{ \begin{array} { l } \alpha _ { j } ^ { ( k ) } = \min _ { 1 \leqslant i \leqslant n } m _ { i , j } ^ { ( k ) } , \\ \beta _ { j } ^ { ( k ) } = \max _ { 1 \leqslant i \leqslant n } m _ { i , j } ^ { ( k ) } . \end{array} \right.$$
In the following four questions, $j$ is a fixed integer in $\llbracket 1 , n \rrbracket$ and $k$ is fixed in $\mathbb { N }$. Prove that there exists a pair $\left( i _ { 1 } , j _ { 1 } \right) \in \llbracket 1 , n \rrbracket ^ { 2 }$ such that $$\beta _ { j } ^ { ( k ) } - \beta _ { j } ^ { ( k + 1 ) } \geqslant m _ { i _ { 1 } , j _ { 1 } } \left( \beta _ { j } ^ { ( k ) } - \alpha _ { j } ^ { ( k ) } \right) .$$

Q24 Matrices Matrix Norm, Convergence, and Inequality View

We now assume that all coefficients $m _ { i , j } ( 1 \leqslant i , j \leqslant n )$ of the stochastic matrix $M$ are strictly positive. We set $\varepsilon = \min _ { 1 \leqslant i , j \leqslant n } m _ { i , j }$. We are interested in the sequence $\left( M ^ { k } \right) _ { k \in \mathbb { N } }$ of powers of $M$. We denote by $m _ { i , j } ^ { ( k ) }$ the coefficient of the matrix $M ^ { k }$ located in row $i$ and column $j$.
For all $j \in \llbracket 1 , n \rrbracket$, we set $$\left\{ \begin{array} { l } \alpha _ { j } ^ { ( k ) } = \min _ { 1 \leqslant i \leqslant n } m _ { i , j } ^ { ( k ) } , \\ \beta _ { j } ^ { ( k ) } = \max _ { 1 \leqslant i \leqslant n } m _ { i , j } ^ { ( k ) } . \end{array} \right.$$
Deduce that $\beta _ { j } ^ { ( k + 1 ) } - \alpha _ { j } ^ { ( k + 1 ) } \leqslant ( 1 - 2 \varepsilon ) \left( \beta _ { j } ^ { ( k ) } - \alpha _ { j } ^ { ( k ) } \right)$.

Q25 Matrices Matrix Power Computation and Application View

We now assume that all coefficients $m _ { i , j } ( 1 \leqslant i , j \leqslant n )$ of the stochastic matrix $M$ are strictly positive. We set $\varepsilon = \min _ { 1 \leqslant i , j \leqslant n } m _ { i , j }$.
Prove that the sequence $\left( M ^ { k } \right)$ converges to a stochastic matrix $B = \left( \begin{array} { l l l } b _ { 1 } & \cdots & b _ { n } \\ b _ { 1 } & \cdots & b _ { n } \\ b _ { 1 } & \cdots & b _ { n } \end{array} \right)$ all of whose rows are equal.

Q29 Matrices Structured Matrix Characterization View

We model the web by a directed graph with $n$ vertices. For every integer $i \in \llbracket 1 , n \rrbracket$, $\lambda _ { i }$ denotes the number of outgoing edges from page $i$. We assume that no page points to itself. A surfer navigates the web in the following way: when on page $i$,

if page $i$ points to other pages, he goes randomly, with equal probability, to one of these pages;
if page $i$ points to no other page, he remains on page $i$.

Verify that the transition matrix associated with this navigation model is the matrix $A = \left( a _ { i , j } \right) _ { 1 \leqslant i , j \leqslant n }$ with $$\left\{ \begin{array} { l } a _ { i , i } = \begin{cases} 1 & \text { if page } i \text { points to no other page } \\ 0 & \text { otherwise } \end{cases} \\ a _ { i , j } = \left\{ \begin{array} { l l } 0 & \text { if } i \nrightarrow j \\ 1 / \lambda _ { i } & \text { if } i \rightarrow j \end{array} \quad \text { for } i \neq j \right. \end{array} \right.$$

Q30 Matrices Structured Matrix Characterization View

We model the web by a directed graph with $n$ vertices. The matrix $A$ is the stochastic matrix described in question 29. We define $$B = ( 1 - \alpha ) A + \frac { \alpha } { n } J _ { n }$$ where $J _ { n }$ is the matrix in $\mathcal { M } _ { n } ( \mathbb { R } )$ whose coefficients are all equal to $1$, $A$ is the stochastic matrix described in question 29 and $\alpha$ is a real number in $] 0,1 [$, called the damping factor.
Show that $B$ is a stochastic matrix whose coefficients are all strictly positive.

Q31 Matrices Linear Transformation and Endomorphism Properties View

We model the web by a directed graph with $n$ vertices. The matrix $A$ is the stochastic matrix described in question 29. We define $$B = ( 1 - \alpha ) A + \frac { \alpha } { n } J _ { n }$$ where $J _ { n }$ is the matrix in $\mathcal { M } _ { n } ( \mathbb { R } )$ whose coefficients are all equal to $1$, $A$ is the stochastic matrix described in question 29 and $\alpha$ is a real number in $] 0,1 [$, called the damping factor.
In the navigation model admitting $B$ as its transition matrix, give the probability of leaving a page containing no links to another page.