For candidates who have followed the specialization course
The two parts are independent.

Part A

A laboratory studies the evolution of a population of parasitic insects on plants. This evolution has two stages: a larval stage and an adult stage which is the only one during which insects can reproduce. Observation of the evolution of this population leads to proposing the following model. Each week:

• Each adult gives birth to 2 larvae then $75\%$ of adults die.
• $25\%$ of larvae die and $50\%$ of larvae become adults.

For all natural integer $n$, we denote $\ell _ { n }$ the number of larvae and $a _ { n }$ the number of adults after $n$ weeks. For all natural integer $n$, we denote $X _ { n }$ the column matrix defined by: $X _ { n } = \binom { \ell _ { n } } { a _ { n } }$

1. Show that, for all natural integer $n$, $X _ { n + 1 } = A X _ { n }$ where $A$ is the matrix: $$A = \left( \begin{array} { c c } 0.25 & 2 \\ 0.5 & 0.25 \end{array} \right)$$
2. We denote $U$ and $V$ the column matrices: $U = \binom { 2 } { 1 }$ and $V = \binom { a } { 1 }$, where $a$ is a real number. a. Show that $A U = 1.25 U$. b. Determine the real number $a$ such that $A V = - 0.75 V$.

In questions 3 and 4, the real number $a$ is fixed so that it is the solution of $A V = - 0.75 V$.

1. It is admitted that there exist two real numbers $\alpha$ and $\beta$ such that: $X _ { 0 } = \alpha U + \beta V$ and $\alpha > 0$. a. Show that, for all natural integer $n$, $X _ { n } = \alpha ( 1.25 ) ^ { n } U + \beta ( - 0.75 ) ^ { n } V$. b. Deduce that for all natural integer $n$: $$\left\{ \begin{array} { l } \ell _ { n } = 2 ( 1.25 ) ^ { n } \left( \alpha - \beta ( - 0.6 ) ^ { n } \right) \\ a _ { n } = ( 1.25 ) ^ { n } \left( \alpha + \beta ( - 0.6 ) ^ { n } \right) . \end{array} \right.$$
2. Show that $\lim _ { n \rightarrow + \infty } \frac { \ell _ { n } } { a _ { n } } = 2$. Interpret this result in the context of the exercise.

Part B

1. We consider the equation $( E ) : 19 x - 6 y = 1$. Determine the number of couples of integers ( $x ; y$ ) solutions of the equation $( E )$ and satisfying $2000 \leqslant x \leqslant 2100$.
2. Let $n$ be a natural integer. Show that the integers ( $2 n + 3$ ) and ( $n + 3$ ) are coprime if and only if $n$ is not a multiple of 3.

Consider the matrix $M = \left( \begin{array} { l l l } 0 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 0 \end{array} \right)$. Let $\left( a _ { n } \right)$ and $\left( b _ { n } \right)$ be two sequences of integers defined by: $$a _ { 1 } = 1 , b _ { 1 } = 0 \text { and for every non-zero natural number } n \begin{cases} a _ { n + 1 } & = a _ { n } + b _ { n } \\ b _ { n + 1 } & = 2 a _ { n } \end{cases}$$

1. Calculate $a _ { 2 } , b _ { 2 } , a _ { 3 }$ and $b _ { 3 }$.
2. Give $M ^ { 2 }$. Show that $M ^ { 2 } = M + 2 I$ where $I = \left( \begin{array} { l l l } 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array} \right)$ denotes the identity matrix of order 3. It is admitted that for every non-zero natural number $n , M ^ { n } = a _ { n } M + b _ { n } I$, where ( $a _ { n }$ ) and ( $b _ { n }$ ) are the previously defined sequences.
3. Let $A = \left( \begin{array} { l l } 1 & 1 \\ 2 & 0 \end{array} \right)$ and for every non-zero natural number $n$, let $X _ { n }$ denote the matrix $\binom { a _ { n } } { b _ { n } }$. Let $P = \left( \begin{array} { c c } 1 & 1 \\ 1 & - 2 \end{array} \right)$. a. Verify that, for every non-zero natural number $n , X _ { n + 1 } = A X _ { n }$. b. Without justification, express, for every integer $n \geqslant 2 , X _ { n }$ in terms of $A ^ { n - 1 }$ and $X _ { 1 }$. c. Justify that $P$ is invertible with inverse $\left( \begin{array} { c c } \frac { 2 } { 3 } & \frac { 1 } { 3 } \\ \frac { 1 } { 3 } & - \frac { 1 } { 3 } \end{array} \right)$. Let $P ^ { - 1 }$ denote this matrix. d. Verify that $P ^ { - 1 } A P$ is a diagonal matrix $D$ which you will specify. e. Prove by induction that for every non-zero natural number $n , A ^ { n } = P D ^ { n } P ^ { - 1 }$. f. It is admitted that for every integer $n \geqslant 1$: $$A ^ { n - 1 } = \left( \begin{array} { l l } \frac { 1 } { 3 } \times 2 ^ { n } + \frac { 1 } { 3 } \times ( - 1 ) ^ { n - 1 } & \frac { 1 } { 3 } \times 2 ^ { n - 1 } + \frac { 1 } { 3 } \times ( - 1 ) ^ { n } \\ \frac { 1 } { 3 } \times 2 ^ { n } - \frac { 2 } { 3 } \times ( - 1 ) ^ { n - 1 } & \frac { 1 } { 3 } \times 2 ^ { n - 1 } - \frac { 2 } { 3 } \times ( - 1 ) ^ { n } \end{array} \right)$$ Deduce that for every integer $n \geqslant 1 , a _ { n } = \frac { 1 } { 3 } \times \left( 2 ^ { n } + ( - 1 ) ^ { n - 1 } \right)$.
4. Prove that, for every natural number $k , 2 ^ { 4 k } - 1 \equiv 0$ modulo 5.
5. Let $n$ be a non-zero natural number and a multiple of 4. a. Show that $3 a _ { n }$ is divisible by 5. b. Deduce that $a _ { n }$ is divisible by 5.

Exercise 4 — Candidates who have followed the specialisation course
For each of the following statements, indicate whether it is true or false, by justifying the answer.
One point is awarded for each correct answer that is properly justified. An unjustified answer is not taken into account. No answer is not penalised.
1. Statement 1: The solutions of the equation $7 x - 12 y = 5$, where $x$ and $y$ are relative integers, are the pairs $( - 1 + 12 k ; - 1 + 7 k )$ where $k$ ranges over the set of relative integers.
2. Statement 2: For all natural number $n$, the remainder of the Euclidean division of $4 + 3 \times 15 ^ { n }$ by 3 is equal to 1.
3. Statement 3: The equation $n \left( 2 n ^ { 2 } - 3 n + 5 \right) = 3$, where $n$ is a natural number, has at least one solution.
4. Let $t$ be a real number. We set $A = \left( \begin{array} { c c } t & 3 \\ 2 t & - t \end{array} \right)$.
Statement 4: There is no value of the real number $t$ for which $A ^ { 2 } = \left( \begin{array} { l l } 1 & 0 \\ 0 & 1 \end{array} \right)$.
5. Consider the matrices $A = \left( \begin{array} { c c c } 0 & 1 & - 1 \\ - 1 & 2 & - 1 \\ 1 & - 1 & 2 \end{array} \right)$ and $I _ { 3 } = \left( \begin{array} { l l l } 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array} \right)$.
Statement 5: For all integer $n \geqslant 2 , A ^ { n } = \left( 2 ^ { n } - 1 \right) A + \left( 2 - 2 ^ { n } \right) I _ { 3 }$.

A matriz $A = \begin{pmatrix} 2 & 1 \\ 3 & 4 \end{pmatrix}$ tem determinante igual a
(A) 2 (B) 3 (C) 5 (D) 8 (E) 11

A student recorded the bimonthly grades of some of his subjects in a table. He observed that the numerical entries in the table formed a $4 \times 4$ matrix, and that he could calculate the annual averages of these subjects using matrix multiplication. All tests had the same weight, and the table he obtained is shown below.

	$1^{st}$ bimonth	$2^{nd}$ bimonth	$3^{rd}$ bimonth	$4^{th}$ bimonth
Mathematics	5.9	6.2	4.5	5.5
Portuguese	6.6	7.1	6.5	8.4
Geography	8.6	6.8	7.8	9.0
History	6.2	5.6	5.9	7.7

To obtain these averages, he multiplied the matrix obtained from the table by
(A) $\left[\frac{1}{2}\quad\frac{1}{2}\quad\frac{1}{2}\quad\frac{1}{2}\right]$

QUESTION 156
The matrix $A = \begin{pmatrix} 2 & 1 \\ 3 & 4 \end{pmatrix}$ has determinant
(A) 2
(B) 3
(C) 4
(D) 5
(E) 6

The matrix $A = \begin{pmatrix} 2 & 1 \\ 3 & 4 \end{pmatrix}$ has determinant:
(A) 2
(B) 3
(C) 4
(D) 5
(E) 6

There exists a real $3 \times 3$ orthogonal matrix with only non-zero entries.

A linear transformation $T : \mathbb{R}^8 \rightarrow \mathbb{R}^8$ is defined on the standard basis $e_1, \ldots, e_8$ by $$\begin{aligned} & T e_j = e_{j+1} \quad j = 1, \ldots, 5 \\ & T e_6 = e_7 \\ & T e_7 = e_6 \\ & T e_8 = e_2 + e_4 + e_6 + e_8. \end{aligned}$$ What is the nullity of $T$?

For any $n \geq 2$ there is an $n \times n$ matrix $A$ with real entries such that $A ^ { 2 } = A$ and trace $( A ) = n + 1$.

The matrix $\left( \begin{array} { c c } \pi & \pi \\ 0 & \frac { 22 } { 7 } \end{array} \right)$ is diagonalizable over $\mathbb { C }$.

(a) Show that there exists a $3 \times 3$ invertible matrix $M \neq I _ { 3 }$ with entries in the field $\mathbb { F } _ { 2 }$ such that $M ^ { 7 } = I _ { 3 }$.
(b) Let $A$ be an $m \times n$ matrix, and $\mathbf { b }$ an $m \times 1$ vector, both with integer entries.

1. Suppose that there exists a prime number $p$ such that the equation $A \mathbf { x } = \mathbf { b }$ seen as an equation over the finite field $\mathbb { F } _ { p }$ has a solution. Then does there exist a solution to $A \mathbf { x } = \mathbf { b }$ over the real numbers?
2. If $A \mathbf { x } = \mathbf { b }$ has a solution over $\mathbb { F } _ { p }$ for every prime $p$, is a real solution guaranteed?

Let $M _ { n } ( \mathbb { C } )$ denote the set of $n \times n$ matrices over $\mathbb { C }$. Think of $M _ { n } ( \mathbb { C } )$ as the $2 n ^ { 2 }$-dimensional Euclidean space $\mathbb { R } ^ { 2 n ^ { 2 } }$. Show that the set of all diagonalizable $n \times n$ matrices is dense in $M _ { n } ( \mathbb { C } )$.

Let $m$ and $n$ be positive integers and $0 \leq k \leq \min\{m,n\}$ an integer. Prove or disprove: The subspace of $M_{m \times n}(\mathbb{C})$ consisting of all matrices of rank equal to $k$ is connected. (You may use the following fact: For $t \geq 2$, $\mathrm{GL}_{t}(\mathbb{C})$ is connected.)

Let $A \in M_{m \times n}(\mathbb{R})$ be of rank $m$. Choose the correct statement(s) from below:
(A) The map $\mathbb{R}^n \longrightarrow \mathbb{R}^m$ given by $v \mapsto Av$ is injective;
(B) There exist matrices $B \in M_m(\mathbb{R})$ and $C \in M_n(\mathbb{R})$ such that $BAC = \left[I_m \mid \mathbf{0}_{n-m}\right]$;
(C) There exist matrices $B \in \mathrm{GL}_m(\mathbb{R})$ and $C \in \mathrm{GL}_n(\mathbb{R})$ such that $BAC = \left[I_m \mid \mathbf{0}_{n-m}\right]$;
(D) For every $(B, C) \in M_m(\mathbb{R}) \times M_n(\mathbb{R})$ such that $BAC = \left[I_m \mid \mathbf{0}_{n-m}\right]$, $C$ is uniquely determined by $B$.

For this question write your answers as a series of four letters (Y for Yes and N for No) in order. Is it possible to find a $2 \times 2$ matrix $M$ for which the equation $M\vec{x} = \vec{p}$ has:
(a) no solutions for some but not all $\vec{p}$; exactly one solution for all other $\vec{p}$?
(b) exactly one solution for some but not all $\vec{p}$; more than one solution for all other $\vec{p}$?
(c) no solutions for some but not all $\vec{p}$; more than one solution for all other $\vec{p}$?
(d) no solutions for some $\vec{p}$, exactly one solution for some $\vec{p}$ and more than one solution for some $\vec{p}$?

Let $a, b \in \mathbb{R}$, and consider the $\mathbb{R}$-linear map $f : \mathbb{C} \longrightarrow \mathbb{C},\ z \mapsto az + b\bar{z}$. Choose the correct statement(s) from below:
(A) $f$ is onto (i.e., surjective) if $ab \neq 0$;
(B) $f$ is one-one (i.e., injective) if $ab \neq 0$;
(C) $f$ is onto if $a^2 \neq b^2$;
(D) if $a^2 = b^2$, $f$ is not one-one.

Let $a_n,\ n \geq 0$ be complex numbers such that $\lim_n a_n = 0$.
(A) Show that $F(z) := \sum_{n \geq 0} a_n z^n$ is a holomorphic function on $\{z \in \mathbb{C} : |z| < 1\}$.
(B) Let $G(z)$ be a meromorphic function on $\{z \in \mathbb{C} : |z| < 2\}$, with a pole at 1. Show that $G \neq F$ on $\{z \in \mathbb{C} : |z| < 1\}$. (Hint: consider the function $(1-z)F(z)$ as $z \longrightarrow 1$.)

Let $V$ be a subspace of the complex vector space $M_n(\mathbb{C})$. Suppose that every non-zero element of $V$ is an invertible matrix. Show that $\dim_{\mathbb{C}} V \leq 1$.

Let $A$ and $B$ be $5 \times 5$ real matrices with $A^{2} = B^{2}$. Which of the following statements is/are correct?
(A) Either $A = B$ or $A = -B$.
(B) $A$ and $B$ have the same eigen spaces.
(C) $A$ and $B$ have the same eigen values.
(D) $A^{13} B^{3} = A^{3} B^{13}$.

Let $A \in M _ { 2 } ( \mathbb { R } )$ be a nonzero matrix. Pick the correct statement(s) from below.
(A) If $A ^ { 2 } = 0$, then $\left( I _ { 2 } - A \right) ^ { 5 } = 0$.
(B) If $A ^ { 2 } = 0$, then ( $I _ { 2 } - A$ ) is invertible.
(C) If $A ^ { 3 } = 0$, then $A ^ { 2 } = 0$.
(D) If $A ^ { 2 } = A ^ { 3 } \neq 0$, then $A$ is invertible.

For $A \in M _ { 3 } ( \mathbb { C } )$, let $W _ { A } = \left\{ B \in M _ { 3 } ( \mathbb { C } ) \mid A B = B A \right\}$. Which of the following is/are true?
(A) For all diagonal $A \in M _ { 3 } ( \mathbb { C } )$, $W _ { A }$ is a linear subspace of $M _ { 3 } ( \mathbb { C } )$ with $\operatorname { dim } _ { \mathbb { C } } W _ { A } \geq 3$.
(B) For all $A \in M _ { 3 } ( \mathbb { C } ) , W _ { A }$ is a linear subspace of $M _ { 3 } ( \mathbb { C } )$ with $\operatorname { dim } _ { \mathbb { C } } W _ { A } > 3$.
(C) There exists $A \in M _ { 3 } ( \mathbb { C } )$ such that $W _ { A }$ is a linear subspace of $M _ { 3 } ( \mathbb { C } )$ with $\operatorname { dim } _ { \mathbb { C } } W _ { A } = 3$.
(D) If $A \in M _ { 3 } ( \mathbb { C } )$ is diagonalizable, then every element of $W _ { A }$ is diagonalizable.

Let $A = \left[\begin{array}{ccc} 1 & 2 & 3 \\ 10 & 20 & 30 \\ 11 & 22 & k \end{array}\right]$ and $\mathbf{v} = \left[\begin{array}{l} x \\ y \\ z \end{array}\right]$, where $k$ is a constant and $x, y, z$ are variables.
Statements
(9) Regardless of the value of $k$, the matrix $A$ is not invertible, i.e., there is no $3 \times 3$ matrix $B$ such that $BA =$ the $3 \times 3$ identity matrix. (10) There is a unique $k$ such that determinant of $A$ is 0. (11) The set of solutions $(x, y, z)$ of the matrix equation $A\mathbf{v} = \left[\begin{array}{l} 0 \\ 0 \\ 0 \end{array}\right]$ is either a line or a plane containing the origin. (12) If the equation $A\mathbf{v} = \left[\begin{array}{c} p \\ q \\ r \end{array}\right]$ has a solution, then it must be true that $q = 10p$.

$M$ is a $3 \times 3$ matrix with integer entries. For $M$ we have (Sum of column 2) $= 4 \times$ (sum of column 1). (Sum of column 3) $= 4 \times$ (sum of column 2). (Sum of row $2) = 6 +$ (sum of row $1$). (Sum of row $3) = 6 +$ (sum of row 2).
Statements
(9) The sum of all the entries in $M$ must be divisible by 21. (10) None of the row sums is divisible by 7. (11) One of the column sums must be divisible by 7. (12) None of the column sums is divisible by 6.

Let $\mathcal { P } _ { n } = \{ f ( x ) \in \mathbb { R } [ x ] \mid \operatorname { deg } f ( x ) \leq n \}$, considered as an ($n + 1$)-dimensional real vector space. Let $T$ be the linear operator $f \mapsto f + \frac { \mathrm { d } f } { \mathrm {~d} x }$ on $\mathcal { P } _ { n }$. Pick the correct statement(s) from below.
(A) $T$ is invertible.
(B) $T$ is diagonalizable.
(C) $T$ is nilpotent.
(D) $( T - I ) ^ { 2 } = ( T - I )$ where $I$ is the identity map.