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]

For a $2 \times 2$ square matrix $A$, the $(i,j)$ component $a_{ij}$ is $$a_{ij} = i - j \quad (i = 1,2,\ j = 1,2)$$ What is the $(2,1)$ component of the matrix $A + A^2 + A^3 + \cdots + A^{2010}$? [4 points]
(1) $-2010$
(2) $-1$
(3) $0$
(4) $1$
(5) $2010$

Let $n \in \mathbb{N}^*$. Let $A \in M_2(\mathbb{R})$ be an antisymmetric matrix. We set $A = \begin{pmatrix} 0 & -\alpha \\ \alpha & 0 \end{pmatrix}$ where $\alpha \in \mathbb{R}$.
Deduce that $E(A)$ exists and that it is a rotation matrix, whose angle we shall specify.

Let $B \in M_3(\mathbb{R})$ be antisymmetric.
Show that $E(B)$ exists and is a rotation matrix. Specify the value of its unoriented angle as a function of $\|B\|_2$.

Let $D \in M_p(\mathbb{K})$ be a diagonal matrix.
Show that $E(D)$ exists and that $E(D) \in GL_p(\mathbb{C})$.

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. 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.
Deduce that $E(A + B)$ exists and that $E(A + B) = E(A)E(B) = E(B)E(A)$.

Let $A \in M_p(\mathbb{C})$ 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 $k$. 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 $A \in M_p(\mathbb{C})$ 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. We denote $P_n(X) = \left(1 + \frac{X}{n}\right)^n$.
Show that $\lim_{n \rightarrow \infty} P_n(B)$ exists.

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.
Deduce that $E(A)$ exists.

For $(A, B) \in \mathcal{M}_d(\mathbb{R})^2$ such that $A$ and $B$ commute, show that $\exp(\mathrm{i}A)\exp(\mathrm{i}B) = \exp(\mathrm{i}(A+B))$.

We fix a matrix $A \in \mathcal{M}_d(\mathbb{R})$.
Show that, for every $n \in \mathbb{N}^*$ and every $R$ large enough, the matrix $$\frac{1}{2\pi} \int_0^{2\pi} \left(R\mathrm{e}^{\mathrm{i}\theta}\right)^n \left(R\mathrm{e}^{\mathrm{i}\theta} I_d - A\right)^{-1} \mathrm{~d}\theta$$ equals $A^{n-1}$.

We fix a matrix $A \in \mathcal{M}_d(\mathbb{R})$. We consider the characteristic polynomial $$\chi_A(X) = \det\left(A - X \cdot I_d\right) = \sum_{k=0}^d a_k X^k$$ Show that for $R$ large enough: $$\chi_A(A) = \frac{1}{2\pi} \int_0^{2\pi} \left(R\mathrm{e}^{\mathrm{i}\theta}\right) \chi_A\left(R\mathrm{e}^{\mathrm{i}\theta}\right) \left(R\mathrm{e}^{\mathrm{i}\theta} I_d - A\right)^{-1} \mathrm{~d}\theta$$

We denote by $\mathrm{GL}_2(\mathbb{Z})$ the set of invertible elements of $\mathcal{M}_2(\mathbb{Z})$. The Dickson polynomials $(D_n)_{n \in \mathbb{N}}$ and $(E_n)_{n \in \mathbb{N}}$ satisfy the recurrences $D_{n+2}(x,a) = x D_{n+1}(x,a) - a D_n(x,a)$ and $E_{n+2}(x,a) = x E_{n+1}(x,a) - a E_n(x,a)$ with $D_0 = 2, D_1(x,a) = x, E_0 = 1, E_1(x,a) = x$.
Let $B \in \mathrm{GL}_2(\mathbb{Z})$. We denote, in this question only, $\sigma = \operatorname{Tr} B$ and $\nu = \det B$. Show for every $n \geqslant 2$, the equality $$B^n = E_{n-1}(\sigma, \nu) \cdot B - \nu E_{n-2}(\sigma, \nu) \cdot I_2$$ where $I_2$ is the identity matrix of order 2.
Establish that $\operatorname{Tr}(B^n) = D_n(\sigma, \nu)$.

We denote by $\mathrm{GL}_2(\mathbb{Z})$ the set of invertible elements of $\mathcal{M}_2(\mathbb{Z})$. For $B \in \mathrm{GL}_2(\mathbb{Z})$ with $\sigma = \operatorname{Tr} B$ and $\nu = \det B$, we have $B^n = E_{n-1}(\sigma,\nu) \cdot B - \nu E_{n-2}(\sigma,\nu) \cdot I_2$ and $\operatorname{Tr}(B^n) = D_n(\sigma,\nu)$.
We denote $A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$, $\tau = \operatorname{Tr} A$, $\delta = \det A$.
Deduce that if $A$ is an $n$-th power ($n \geqslant 2$) in $\mathrm{GL}_2(\mathbb{Z})$, then there exist $\sigma \in \mathbb{Z}$ and $\nu \in \{-1,1\}$ such that
i. $E_{n-1}(\sigma, \nu)$ divides $b$, $c$ and $a - d$. One will briefly justify that $E_{n-1}(\sigma, \nu)$ is indeed an integer.
ii. $\tau = D_n(\sigma, \nu)$ and $\delta = \nu^n$.

We denote by $\mathrm{GL}_2(\mathbb{Z})$ the set of invertible elements of $\mathcal{M}_2(\mathbb{Z})$. We denote $A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$, $\tau = \operatorname{Tr} A$, $\delta = \det A$.
Let $A$ be an element of $\mathrm{GL}_2(\mathbb{Z})$ for which there exist $\sigma \in \mathbb{Z}$ and $\nu \in \{-1,1\}$ satisfying: i. $E_{n-1}(\sigma, \nu)$ divides $b$, $c$ and $a - d$. ii. $\tau = D_n(\sigma, \nu)$ and $\delta = \nu^n$.
For simplicity, we denote $p = E_{n-1}(\sigma, \nu)$. We then define a matrix $B = \begin{pmatrix} r & s \\ t & u \end{pmatrix}$ with $$r = \frac{1}{2}\left(\sigma + \frac{a-d}{p}\right) \quad s = \frac{b}{p} \quad t = \frac{c}{p} \quad u = \frac{1}{2}\left(\sigma - \frac{a-d}{p}\right)$$
a) By introducing a complex root of the polynomial $X^2 - \sigma X + \nu$ and using the relation $D_n(x + a/x, a) = x^n + a^n/x^n$, show that $$\tau^2 - 4\delta = p^2(\sigma^2 - 4\nu) \quad \text{then} \quad ru - st = \nu$$ Deduce that $B$ belongs to $\mathrm{GL}_2(\mathbb{Z})$.
b) Show that $A = B^n$.

We denote by $\mathrm{GL}_2(\mathbb{Z})$ the set of invertible elements of $\mathcal{M}_2(\mathbb{Z})$. The necessary and sufficient condition for $A \in \mathrm{GL}_2(\mathbb{Z})$ to be an $n$-th power is the existence of $\sigma \in \mathbb{Z}$ and $\nu \in \{-1,1\}$ such that $E_{n-1}(\sigma,\nu)$ divides $b$, $c$ and $a-d$, and $\tau = D_n(\sigma,\nu)$, $\delta = \nu^n$.
Show that the matrix $A = \begin{pmatrix} 7 & 10 \\ 5 & 7 \end{pmatrix}$ is a cube in $\mathrm{GL}_2(\mathbb{Z})$ and determine a matrix $B \in \mathrm{GL}_2(\mathbb{Z})$ such that $B^3 = A$.

Calculate the exponential of the matrix $M_{p,q,r}$, where $$M_{p,q,r} = \begin{pmatrix} 0 & p & r \\ 0 & 0 & q \\ 0 & 0 & 0 \end{pmatrix}.$$

For $m \in \mathbb{R}$, we denote by $\mathcal{M}$ the matrix $$\mathcal{M} = \left(\begin{array}{ccc} 0 & -m & 0 \\ m & 0 & 0 \\ 0 & 0 & 0 \end{array}\right)$$
For $t \in \mathbb{R}$ and $n \in \mathbb{N}^{*}$, we introduce the matrix $$F_{n}(t) = I_{3} + \sum_{k=1}^{n} \frac{t^{k} \mathcal{M}^{k}}{k!} \in \mathcal{M}_{n}(\mathbb{K})$$
Show that the sequence $\left(F_{n}(t)\right)_{n \in \mathbb{N}}$ converges to a limit denoted $F(t) \in \mathcal{M}_{3}(\mathbb{R})$. Express $F(t)$ in terms of $R(\theta)$, where $\theta$ depends on $m$ and $t$.

Let $p \in ]0,1[$. Let $X_1, \ldots, X_n$ be mutually independent random variables following the same Bernoulli distribution with parameter $p$. Let $S = X_1 + \ldots + X_n$, $U(\omega) = (X_1(\omega), \ldots, X_n(\omega))^T$ and $M(\omega) = U(\omega)\,{}^t(U(\omega))$.
Express $M^k$ in terms of $S$ and $M$.
What is the probability that the sequence of matrices $(M^k)_{k \in \mathbb{N}}$ is convergent?
Show that, in this case, the limit is a projection matrix.