(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?

Show that $\mathrm{O}(n)$ is a compact subset of $\mathcal{M}_n(\mathbb{R})$.

Let $\varphi$ be the map from $\mathrm{O}(n) \times \mathcal{S}_n^{++}(\mathbb{R})$ to $\mathrm{GL}_n(\mathbb{R})$ defined by $\varphi(O, S) = OS$ for every pair $(O, S)$ of $\mathrm{O}(n) \times \mathcal{S}_n^{++}(\mathbb{R})$.
Show that $\varphi$ is bijective, continuous, and that its inverse is continuous.

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 $\alpha \in \mathbb{R}$.
Determine a number $\beta_n \in \mathbb{R}_+^*$ such that $$\frac{1}{\beta_n}\left(I_2 + \frac{1}{n}A\right) \in SO_2(\mathbb{R})$$

Let $\mathcal{U}_q$ be the set of endomorphisms $\phi \in \mathcal{L}(V)$ that are compatible with $P_a$. Show that $\mathcal{U}_q$ is a subalgebra of $\mathcal{L}(V)$.

Calculate the product ${}^t\left(R_{p,q}(\theta)\right) R_{p,q}(\theta)$. What property of $R_{p,q}(\theta)$ is recognized?

We consider the set of square matrices of size 3 that are strictly upper triangular: $$\mathbf{L} = \left\{ M_{p,q,r} \mid (p,q,r) \in \mathbf{R}^3 \right\} \quad \text{where} \quad M_{p,q,r} = \begin{pmatrix} 0 & p & r \\ 0 & 0 & q \\ 0 & 0 & 0 \end{pmatrix},$$ and $\mathbf{H} = \{I_3 + M \mid M \in \mathbf{L}\}$, with the group law $M * N = M + N + \frac{1}{2}[M,N]$ on $\mathbf{L}$.
Show that $\mathbf{H}$ equipped with the usual product of matrices is a subgroup of $\mathrm{SL}_3(\mathbf{R})$ and that $$\exp : (\mathbf{L}, *) \rightarrow (\mathbf{H}, \times)$$ is a group isomorphism.

We consider two matrices $A$ and $B$ of $\mathcal{M}_d(\mathbf{R})$ that commute with $[A,B]$. We denote $\mathcal{L} = \operatorname{Vect}(A, B, [A,B])$.
(a) If $M, N \in \mathcal{L}$, show that $[M,N]$ commutes with $M$ and $N$.
(b) Let $G = \{\exp(M) \mid M \in \mathcal{L}\}$. Show that $(G, \times)$ is a group and that the map $$\Phi : \mathbf{H} \rightarrow G, \quad \exp(M_{p,q,r}) \mapsto \exp(pA + qB + r[A,B])$$ is a group homomorphism.

Does the matrix $\Delta _ { p + 1 }$ belong to the set $O ( 1 , p )$ ? to the set $O ^ { + } ( 1 , p )$ ?

Show that, for every matrix $L$ element of $O ( 1 , p )$, its transpose ${ } ^ { t } L$ is also an element of $O ( 1 , p )$.

Show that the sets $O ( 1 , p ) , O ^ { + } ( 1 , p )$ and $O ^ { - } ( 1 , p )$ of $\mathcal { M } _ { p + 1 } ( \mathbb { R } )$ are closed.

(a) Show that $O _ { n } ( \mathbb { R } )$ is a subgroup of the group $\mathrm { GL } _ { n } ( \mathbb { R } )$ of invertible matrices.
(b) Show that $O _ { n } ( \mathbb { R } )$ is a compact subset of $\mathcal { M } _ { n } ( \mathbb { R } )$.

Give two definitions of a vector isometry of $\mathbb{R}^n$ and prove their equivalence.

Prove that $\mathcal{P}_n = \mathcal{X}_n \cap \mathrm{O}_n(\mathbb{R})$ and determine its cardinality.

We are given a matrix $M$ of $\mathrm{GL}_n(\mathbb{R})$ whose coefficients are all natural integers and such that the set formed by all coefficients of all successive powers of $M$ is finite.
Prove that $M^{-1}$ has coefficients in $\mathbb{N}$ and deduce that $M$ is a permutation matrix. What can be said of the converse?

Two simplexes $\mathcal{S}$ and $\mathcal{S}'$ in $\mathbb{R}^n$ are called equivalent if there exist an enumeration of the vertices $s_0, s_1, \ldots, s_n$ of $\mathcal{S}$, and $s_0', s_1', \ldots, s_n'$ of $\mathcal{S}'$, and a matrix $A$ in $\mathrm{GL}_n(\mathbb{Z})$ such that $A(s_i - s_0) = s_i' - s_0'$ for all $i = 1, \ldots, n$.
Show that an integer simplex $\mathcal{S}$ is equivalent to an integer simplex contained in the cube $[0, n!\operatorname{Vol}(\mathcal{S})]^n$.
One may use question 6 for a suitably chosen matrix $M$.

Show that if $M \in M _ { n } ( \mathbb { R } )$ is antisymmetric, the matrix $$O = \left( I _ { n } + M \right) ^ { - 1 } \left( I _ { n } - M \right)$$ is orthogonal.

We prove Broyden's theorem by induction on the dimension. We assume $| \alpha | < 1$ and we introduce the matrices $$Q _ { - } = P - \frac { r { } ^ { t } q } { \alpha - 1 } , \quad Q _ { + } = P - \frac { r { } ^ { t } q } { \alpha + 1 }$$ where $O = \left( \begin{array} { l l } P & r \\ { } ^ { t } q & \alpha \end{array} \right)$ with $P \in M _ { n - 1 } ( \mathbb { R } )$, $r , q \in \mathbb { R } ^ { n - 1 }$, $\alpha \in \mathbb { R }$.
Show that the matrices $Q _ { + }$ and $Q _ { - }$ are orthogonal.

We fix a polynomial $f \in \mathbb{C}[X]$ of degree $n \geq 1$. We consider a complex number $z \in \overline{\mathbb{D}}$ and we define the matrices $M \in \mathcal{M}_{n+1}(\mathbb{C})$ and $P \in \mathcal{M}_{n+1,1}(\mathbb{C})$ by $$M = \left(\begin{array}{cccccc} z & 0 & 0 & \ldots & 0 & \sqrt{1-|z|^2} \\ \sqrt{1-|z|^2} & 0 & 0 & \ldots & 0 & -\bar{z} \\ 0 & & & & & 0 \\ 0 & & & & & 0 \\ \vdots & & I_{n-1} & & & \vdots \\ 0 & & & & & 0 \end{array}\right)$$ and $$P = \left(\begin{array}{c} 1 \\ 0 \\ \vdots \\ 0 \end{array}\right)$$ Show that $M$ is a unitary matrix.

Show that the inverse of a symplectic matrix is a symplectic matrix.

Show that the product of two symplectic matrices is a symplectic matrix. Is the set $\mathrm{Sp}_{2n}(\mathbb{R})$ a vector subspace of $\mathcal{M}_{2n}(\mathbb{R})$?

In this part, $\mathbf{K} = \mathbf{R}$. For every natural integer $n$, $n \geq 2$, we introduce the set, called the special linear group: $$\mathrm{SL}_n(\mathbf{R}) = \left\{ M \in \mathcal{M}_n(\mathbf{R}) \mid \operatorname{det}(M) = 1 \right\}.$$ If $G$ is a closed subgroup of $\mathrm{GL}_n(\mathbf{R})$, we introduce its Lie algebra: $$\mathcal{A}_G = \left\{ M \in \mathcal{M}_n(\mathbf{R}) \mid \forall t \in \mathbf{R} \quad e^{tM} \in G \right\}.$$
$\mathbf{9}$ ▷ Determine $\mathcal{A}_G$ when $G = \mathrm{SL}_n(\mathbf{R})$.

In this part, $\mathbf{K} = \mathbf{R}$. If $G$ is a closed subgroup of $\mathrm{GL}_n(\mathbf{R})$, we introduce its Lie algebra: $$\mathcal{A}_G = \left\{ M \in \mathcal{M}_n(\mathbf{R}) \mid \forall t \in \mathbf{R} \quad e^{tM} \in G \right\}.$$
$\mathbf{10}$ ▷ If $G = \mathrm{O}_n(\mathbf{R})$, show that $\mathcal{A}_G = \mathcal{A}_n(\mathbf{R})$, the set of antisymmetric matrices.

In this part, $\mathbf{K} = \mathbf{R}$. If $G$ is a closed subgroup of $\mathrm{GL}_n(\mathbf{R})$, we introduce its Lie algebra: $$\mathcal{A}_G = \left\{ M \in \mathcal{M}_n(\mathbf{R}) \mid \forall t \in \mathbf{R} \quad e^{tM} \in G \right\}.$$ In questions 11) to 14), $G$ is an arbitrary closed subgroup of $\mathrm{GL}_n(\mathbf{R})$.
$\mathbf{11}$ ▷ Using part 2, show that $\mathcal{A}_G$ is a vector subspace of $\mathcal{M}_n(\mathbf{R})$.

In this part, $\mathbf{K} = \mathbf{R}$. If $G$ is a closed subgroup of $\mathrm{GL}_n(\mathbf{R})$, we introduce its Lie algebra: $$\mathcal{A}_G = \left\{ M \in \mathcal{M}_n(\mathbf{R}) \mid \forall t \in \mathbf{R} \quad e^{tM} \in G \right\}.$$ In questions 11) to 14), $G$ is an arbitrary closed subgroup of $\mathrm{GL}_n(\mathbf{R})$.
$\mathbf{12}$ ▷ Let $A \in \mathcal{A}_G$ and $B \in \mathcal{A}_G$. Show that the application $$\begin{aligned} u : \mathbf{R} & \longrightarrow \mathcal{M}_n(\mathbf{R}) \\ t & \longmapsto u(t) = e^{tA} \cdot B \cdot e^{-tA} \end{aligned}$$ takes values in $\mathcal{A}_G$.