Exercise 5 (5 points) — Candidates who have followed the speciality course
In a given territory, we are interested in the coupled evolution of two species: the buzzards (the predators) and the voles (the prey). Scientists model, for all natural integer $n$, this evolution by:
$$\left\{ \begin{aligned} b _ { 0 } & = 1000 \\ c _ { 0 } & = 1500 \\ b _ { n + 1 } & = 0,3 b _ { n } + 0,5 c _ { n } \\ c _ { n + 1 } & = - 0,5 b _ { n } + 1,3 c _ { n } \end{aligned} \right.$$
where $b _ { n }$ represents approximately the number of buzzards and $c _ { n }$ the approximate number of voles on June 1st of the year $2000 + n$ (where $n$ denotes a natural integer).

1. We denote $A$ the matrix $\left( \begin{array} { c c } 0,3 & 0,5 \\ - 0,5 & 1,3 \end{array} \right)$ and, for all natural integer $n , U _ { n }$ the column matrix $\binom { b _ { n } } { c _ { n } }$. a. Verify that $U _ { 1 } = \binom { 1050 } { 1450 }$ and calculate $U _ { 2 }$. b. Verify that, for all natural integer $n , U _ { n + 1 } = A U _ { n }$.
2. We are given the matrices $P = \left( \begin{array} { l l } 1 & 0 \\ 1 & 1 \end{array} \right) , T = \left( \begin{array} { c c } 0,8 & 0,5 \\ 0 & 0,8 \end{array} \right)$ and $I = \left( \begin{array} { l l } 1 & 0 \\ 0 & 1 \end{array} \right)$. We admit that $P$ has as its inverse a matrix $Q$ of the form $\left( \begin{array} { l l } 1 & 0 \\ a & 1 \end{array} \right)$ where $a$ is a real number. a. Determine the value of $a$ by justifying. b. We admit that $A = P T Q$. Prove that, for all non-zero integer $n$, we have $$A ^ { n } = P T ^ { n } Q .$$ c. Prove using a proof by induction that, for all non-zero integer $n$, $$T ^ { n } = \left( \begin{array} { c c } 0,8 ^ { n } & 0,5 n \times 0,8 ^ { n - 1 } \\ 0 & 0,8 ^ { n } \end{array} \right)$$
3. Lucie executes the algorithm below and obtains as output $N = 40$. What conclusion can Lucie state for the buzzards and the voles? \begin{verbatim} Initialization: N takes the value 0 B takes the value 1000 C takes the value 1500 Processing: While B > 2 or C > 2 N takes the value N + 1 R takes the value B B takes the value 0,3 R + 0,5 C C takes the value -0,5 R + 1,3 C End While Output: Display N \end{verbatim}
4. We admit that, for all non-zero natural integer $n$, we have $$U _ { n } = \binom { 1000 \times 0,8^n + 500 n \times 0,8^{n-1} }{ \text{(expression for } c_n\text{)}}$$

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.

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 $q = 10p$.

Let $A = \left[ \begin{array} { c c c } 1 & 2 & 3 \\ 10 & 20 & 31 \\ 11 & 22 & k \end{array} \right]$ 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
(37) Regardless of the value of $k$, the matrix $A$ is not invertible, i.e., there is no $3 \times 3$ matrix $B$ such that $B A =$ the $3 \times 3$ identity matrix. (38) There is a unique $k$ such that determinant of $A$ is 0. (39) 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. (40) 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 $q = 10 p$.

A $2 \times 2$ square matrix $A$ has the sum of all components equal to 0 and satisfies
$$A ^ { 2 } + A ^ { 3 } = - 3 A - 3 E$$
Find the sum of all components of the matrix $A ^ { 4 } + A ^ { 5 }$. (Here, $E$ is the identity matrix.) [4 points]

Two square matrices $A$ and $B$ satisfy $$A + B = ( B A ) ^ { 2 } , \quad A B A = B + E$$ Among the following statements, which are correct? (Here, $E$ is the identity matrix.) [4 points]
Statements ㄱ. $A = B ^ { 2 }$ ㄴ. $B ^ { - 1 } = A ^ { 2 } + E$ ㄷ. $A ^ { 5 } - B ^ { 5 } = A + B$
(1) ㄱ
(2) ㄴ
(3) ㄱ, ㄷ
(4) ㄴ, ㄷ
(5) ㄱ, ㄴ, ㄷ

Let $D \in M_p(\mathbb{K})$ be a diagonal matrix.
Show that there exists a polynomial $Q \in \mathbb{C}[X]$ such that $Q(D) = E(D)$.

Let $D \in M_p(\mathbb{K})$ be a diagonal matrix. Let $(\Delta, +)$ be the additive subgroup of $M_p(\mathbb{R})$ formed by diagonal matrices.
Show that $E$ defines a group morphism from $(\Delta, +)$ to $(GL_p(\mathbb{R}), \times)$.

Let $A \in M_p(\mathbb{K})$ be a diagonalizable matrix.
Show that $E(A)$ exists.

Let $A \in M_p(\mathbb{K})$ be a diagonalizable matrix.
Show that $\det(E(A)) = e^{\operatorname{tr}(A)}$.

Let $A \in M_p(\mathbb{K})$ be a diagonalizable matrix. Let $x \in \mathbb{C}$.
Show that $E\left(xI_p + A\right)$ exists and that $$E\left(xI_p + A\right) = e^x E(A)$$

Let $A, B \in M_p(\mathbb{K})$ be two diagonalizable matrices. We assume that $A$ and $B$ commute.
Show that there exists $P \in GL_p(\mathbb{C})$ such that $P^{-1}AP$ and $P^{-1}BP$ are diagonal.
(We shall study the restrictions of $u_B$ to the eigenspaces of $u_A$.)

Let $A, B \in M_p(\mathbb{K})$ be two diagonalizable matrices. We assume that $A$ and $B$ commute.
Deduce that $E(A + B)$ exists and that $E(A + B) = E(A)E(B) = E(B)E(A)$.

Let $A \in M_p(\mathbb{C})$ and $k \in \mathbb{N}^*$ such that $A^k = 0$ and $A^{k-1} \neq 0$ (we say that $A$ is nilpotent of order $k$).
Show that, for every integer $j$ such that $1 \leqslant j \leqslant k$, $\operatorname{Ker} A^{j-1}$ is strictly included in $\operatorname{Ker} A^j$.

Let $A \in M_p(\mathbb{C})$ and $k \in \mathbb{N}^*$ such that $A^k = 0$ and $A^{k-1} \neq 0$ (we say that $A$ is nilpotent of order $k$).
Deduce that $k \leqslant p$.

Let $A \in M_p(\mathbb{C})$ be nilpotent of order $k$.
Show that $E(A)$ exists. Propose a Maple or Mathematica procedure taking as input a strictly upper triangular matrix $A$ and returning the value of $E(A)$.

Let $A \in M_p(\mathbb{C})$ be nilpotent of order $k$.
Show that there exists a polynomial $Q \in \mathbb{C}[X]$ such that $Q(A) = E(A)$.

Let $A \in M_p(\mathbb{C})$ be nilpotent of order $k$. Let $B \in M_p(\mathbb{C})$. We assume that $A$ and $B$ commute and that $E(B)$ exists.
We admit that, for every integer $i$ between 1 and $p$, $$\lim_{n \rightarrow \infty} \left(I_p + \frac{1}{n}B\right)^n = \lim_{n \rightarrow \infty} \left(I_p + \frac{1}{n}B\right)^{n-i}$$
Show that $E(A + B)$ exists and that $E(A + B) = E(A)E(B)$.

Let $A \in M_p(\mathbb{C})$ be nilpotent of order $k$. Let $x \in \mathbb{C}$.
Show that $E\left(xI_p + A\right)$ exists and that $E\left(xI_p + A\right) = e^x E(A)$.

Let $A \in M_p(\mathbb{C})$ and $n \in \mathbb{N}^*$. We denote $$P_n(X) = \left(1 + \frac{X}{n}\right)^n \in \mathbb{C}[X]$$ and $\chi_A$ the characteristic polynomial of $A$, with the Euclidean division $P_n = Q_n \chi_A + R_n$.
Show that $E(A)$ exists if and only if $\lim_{n \rightarrow \infty} R_n(A)$ exists.

Let $A \in M_p(\mathbb{C})$. Let $k \in \mathbb{N}^*$ and $\lambda_1, \lambda_2, \ldots, \lambda_k$ be the roots of $\chi_A$ pairwise distinct, whose respective multiplicities we denote by $n_1, n_2, \ldots, n_k$.
For every integer $q$ between 1 and $p$, we denote by $J_q$ the matrix of $M_q(\mathbb{C})$ whose coefficients are all zero except those located just above the diagonal which equal 1.
Show that, for every $x \in \mathbb{C}$, for every integer $q$ between 1 and $p$, the family $\left\{\left(xI_q + J_q\right)^i,\ 0 \leqslant i \leqslant q-1\right\}$ is free.

Let $A \in M_p(\mathbb{C})$. Let $k \in \mathbb{N}^*$ and $\lambda_1, \lambda_2, \ldots, \lambda_k$ be the roots of $\chi_A$ pairwise distinct, with respective multiplicities $n_1, n_2, \ldots, n_k$. For every integer $q$ between 1 and $p$, $J_q$ denotes the matrix of $M_q(\mathbb{C})$ whose coefficients are all zero except those just above the diagonal which equal 1.
Let $B = \operatorname{diag}\left\{\lambda_1 I_{n_1} + J_{n_1}, \ldots, \lambda_k I_{n_k} + J_{n_k}\right\}$ be the block diagonal matrix defined by $$B = \begin{pmatrix} \lambda_1 I_{n_1} + J_{n_1} & 0 & \ldots & 0 \\ 0 & \lambda_2 I_{n_2} + J_{n_2} & \cdots & 0 \\ \vdots & \ddots & \ddots & 0 \\ 0 & \ldots & 0 & \lambda_k I_{n_k} + J_{n_k} \end{pmatrix}$$
Show that $\chi_B = \chi_A$.

Let $A \in M_p(\mathbb{C})$ and $B = \operatorname{diag}\left\{\lambda_1 I_{n_1} + J_{n_1}, \ldots, \lambda_k I_{n_k} + J_{n_k}\right\}$ as defined in V.A.4. Let $i$ be an integer $\geqslant 1$.
Show that $$B^i = \begin{pmatrix} \left(\lambda_1 I_{n_1} + J_{n_1}\right)^i & 0 & \cdots & 0 \\ 0 & \left(\lambda_2 I_{n_2} + J_{n_2}\right)^i & \cdots & 0 \\ \vdots & \ddots & \ddots & 0 \\ 0 & \cdots & 0 & \left(\lambda_k I_{n_k} + J_{n_k}\right)^i \end{pmatrix}$$

Let $A \in M_p(\mathbb{C})$ and $B = \operatorname{diag}\left\{\lambda_1 I_{n_1} + J_{n_1}, \ldots, \lambda_k I_{n_k} + J_{n_k}\right\}$ as defined in V.A.4. Let $P$ be a non-zero annihilating polynomial of the matrix $B$.
a) Show that the degree of $P$ is $\geqslant p$.
b) Deduce that the family $\left\{B^i,\ 0 \leqslant i \leqslant p-1\right\}$ is free.

Let $A \in M_p(\mathbb{C})$ and $B = \operatorname{diag}\left\{\lambda_1 I_{n_1} + J_{n_1}, \ldots, \lambda_k I_{n_k} + J_{n_k}\right\}$ as defined in V.A.4. We denote $P_n(X) = \left(1 + \frac{X}{n}\right)^n$.
Show that $\lim_{n \rightarrow \infty} P_n(B)$ exists.