Let $A$ be a matrix of $\mathbf{GL}_n$ similar to its inverse. We admit that $A$ is similar to a block diagonal matrix of the form $$A' = \left(\begin{array}{cccc} J_{n_1}(\lambda_1) & 0 & \ldots & 0 \\ 0 & J_{n_2}(\lambda_2) & \ddots & \vdots \\ \vdots & \ddots & \ddots & 0 \\ 0 & \ldots & 0 & J_{n_r}(\lambda_r) \end{array}\right)$$ where the $\lambda_i$ are the eigenvalues of $A$ (not necessarily distinct) and $r$ as well as the $n_i$, $1 \leq i \leq r$, are nonzero natural integers. Moreover the matrix $A'$ is unique up to the order of the blocks. Prove that $A^{-1}$ is similar to $\left(\begin{array}{cccc} J_{n_1}\left(\frac{1}{\lambda_1}\right) & 0 & \cdots & 0 \\ 0 & J_{n_2}\left(\frac{1}{\lambda_2}\right) & \ddots & \vdots \\ \vdots & \ddots & \ddots & 0 \\ 0 & \cdots & 0 & J_{n_r}\left(\frac{1}{\lambda_r}\right) \end{array}\right)$.

Deduce that the matrix $J(p)$ is invertible if and only if $p$ has no stable root.

In this part, we assume that $n$ is a power of 2: we write $n = 2 ^ { k }$ with $k \in \mathbf { N } ^ { \star }$. Deduce that there exists a vector subspace $F$ of dimension $k$ of $\mathbf { R } ^ { n }$ such that: $$\forall x \in F , \quad \alpha _ { 1 } \sqrt { n } \| x \| _ { 2 } ^ { \mathbf { R } ^ { n } } \leq \| x \| _ { 1 } ^ { \mathbf { R } ^ { n } } \leq \beta _ { 1 } \sqrt { n } \| x \| _ { 2 } ^ { \mathbf { R } ^ { n } } .$$ By ordering the $n$ elements of $\{ - 1,1 \} ^ { k }$ arbitrarily, you may use the map $T$ defined on $\mathbf { R } ^ { k }$ by $T \left( a _ { 1 } , \ldots , a _ { k } \right) = \left( \sum _ { i = 1 } ^ { k } a _ { i } \varepsilon _ { i } \right) _ { \left( \varepsilon _ { 1 } , \ldots , \varepsilon _ { k } \right) \in \{ - 1,1 \} ^ { k } }$.

Let $A$ be a matrix of $\mathbf{GL}_n$ similar to its inverse. We admit that $A$ is similar to a block diagonal matrix of the form $$A' = \left(\begin{array}{cccc} J_{n_1}(\lambda_1) & 0 & \cdots & 0 \\ 0 & J_{n_2}(\lambda_2) & \ddots & \vdots \\ \vdots & \ddots & \ddots & 0 \\ 0 & \cdots & 0 & J_{n_r}(\lambda_r) \end{array}\right)$$ where the $\lambda_i$ are the eigenvalues of $A$ (not necessarily distinct) and $r$ as well as the $n_i$, $1 \leq i \leq r$, are nonzero natural integers. Moreover the matrix $A'$ is unique up to the order of the blocks. One may also use that $J_n(-1)$ is a product of two symmetry matrices.
Using the results established in the previous parts, prove that $A$ is a product of two symmetry matrices.

Let $A$ be a matrix of $\mathbf{GL}_n$ similar to its inverse. We admit that $A$ is similar to a block diagonal matrix of the form $$A' = \left(\begin{array}{cccc} J_{n_1}(\lambda_1) & 0 & \ldots & 0 \\ 0 & J_{n_2}(\lambda_2) & \ddots & \vdots \\ \vdots & \ddots & \ddots & 0 \\ 0 & \ldots & 0 & J_{n_r}(\lambda_r) \end{array}\right)$$ where the $\lambda_i$ are the eigenvalues of $A$ (not necessarily distinct) and $r$ as well as the $n_i$, $1 \leq i \leq r$, are nonzero natural integers. Moreover the matrix $A'$ is unique up to the order of the blocks. Using the results established in the previous parts, prove that $A$ is a product of two symmetry matrices.

We assume in this part that all roots of $p$ are stable and have multiplicity 1 and we denote by $h = Xp'$ (where $p'$ is the derivative polynomial of $p$) and $h_0$ the reciprocal polynomial of $h$. We recall that, according to question 3, there exists a real number $\lambda \in \{-1, 1\}$ such that $p = \lambda p_0$.
Show that $J(h)$ is invertible.

In this subsection, we assume that $J_n = J_n^{(1)}$, the matrix introduced in subsection A-IV. We adopt the following convention: for all $x = (x_1, \ldots, x_n) \in \Lambda_n$, we denote $x_{n+1} = x_1$ and $x_0 = x_n$.
We set $A = \begin{pmatrix} \mathrm{e}^{\beta - h} & \mathrm{e}^{-\beta - h} \\ \mathrm{e}^{-\beta + h} & \mathrm{e}^{\beta + h} \end{pmatrix}$.
Show that $Z_n(h) = \operatorname{tr}(A^n)$, where $\operatorname{tr}$ denotes the trace of a square matrix.

In this fourth part, $A \in \mathcal{S}_n(\mathbb{R})$ is a symmetric matrix whose eigenvalues are denoted $\lambda_1 \leqslant \lambda_2 \leqslant \cdots \leqslant \lambda_n$. We consider an arbitrary orthonormal basis $\left(\mathbf{u}_1, \ldots, \mathbf{u}_n\right)$. Let $\mathbf{U}$ be a random variable taking values in the finite set $\left\{\mathbf{u}_1, \ldots, \mathbf{u}_n\right\}$, following the uniform distribution on this set. We consider the random variable $B = A + \mathbf{U}\mathbf{U}^T$. Prove that there exists $x \in \mathbb{R}$ such that $\mathbb{E}\left[\chi_B(x)\right] \neq 0$.

As in the third part, we suppose that $B = A + \mathbf{u}\mathbf{u}^T$ with $A \in \mathcal{S}_n(\mathbb{R})$ a symmetric matrix, and $\mathbf{u} \in \mathbb{R}^n$ a vector such that $\|\mathbf{u}\| = 1$. We denote by $\lambda_1 \leqslant \lambda_2 \leqslant \cdots \leqslant \lambda_n$ the eigenvalues of $A$ and $\mu_1 \leqslant \mu_2 \leqslant \cdots \leqslant \mu_n$ those of $B$. We admit that $$\lambda_1 \leqslant \mu_1 \leqslant \lambda_2 \leqslant \mu_2 \leqslant \cdots \leqslant \lambda_n \leqslant \mu_n.$$ We further suppose that there exists an integer $m \in \{1, 2, \ldots, n-1\}$ such that the eigenvalues of $A$ satisfy $$0 = \lambda_1 = \lambda_2 = \cdots = \lambda_m < \lambda_{m+1} \leqslant \cdots \leqslant \lambda_n.$$ Let $\varepsilon \in ]0, \lambda_{m+1}[$.
Justify that $(A - \varepsilon \mathbb{I}_n)$ is invertible.

As in the third part, we assume that $B = A + \mathbf{u u}^T$ with $A \in \mathcal{S}_n(\mathbb{R})$ a symmetric matrix, and $\mathbf{u} \in \mathbb{R}^n$ a vector such that $\|\mathbf{u}\| = 1$. We denote by $\lambda_1 \leqslant \lambda_2 \leqslant \cdots \leqslant \lambda_n$ the eigenvalues of $A$ and $\mu_1 \leqslant \mu_2 \leqslant \cdots \leqslant \mu_n$ those of $B$. We admit that $$\lambda_1 \leqslant \mu_1 \leqslant \lambda_2 \leqslant \mu_2 \leqslant \cdots \leqslant \lambda_n \leqslant \mu_n.$$ We further assume that there exists an integer $m \in \{1,2,\ldots,n-1\}$ such that the eigenvalues of $A$ satisfy $$0 = \lambda_1 = \lambda_2 = \cdots = \lambda_m < \lambda_{m+1} \leqslant \cdots \leqslant \lambda_n.$$ Let $\varepsilon \in ]0, \lambda_{m+1}[$. Justify that $(A - \varepsilon \mathbb{I}_n)$ is invertible.

As in the third part, we suppose that $B = A + \mathbf{u}\mathbf{u}^T$ with $A \in \mathcal{S}_n(\mathbb{R})$ a symmetric matrix, and $\mathbf{u} \in \mathbb{R}^n$ a vector such that $\|\mathbf{u}\| = 1$. We denote by $\lambda_1 \leqslant \lambda_2 \leqslant \cdots \leqslant \lambda_n$ the eigenvalues of $A$ and $\mu_1 \leqslant \mu_2 \leqslant \cdots \leqslant \mu_n$ those of $B$. We admit that $$\lambda_1 \leqslant \mu_1 \leqslant \lambda_2 \leqslant \mu_2 \leqslant \cdots \leqslant \lambda_n \leqslant \mu_n.$$ We further suppose that there exists an integer $m \in \{1, 2, \ldots, n-1\}$ such that the eigenvalues of $A$ satisfy $$0 = \lambda_1 = \lambda_2 = \cdots = \lambda_m < \lambda_{m+1} \leqslant \cdots \leqslant \lambda_n.$$ Let $\varepsilon \in ]0, \lambda_{m+1}[$. We suppose that $\left\langle \mathbf{u}, \left(A - \varepsilon \mathbb{I}_n\right)^{-1}\mathbf{u}\right\rangle < -1$.
Show that $(B - \varepsilon \mathbb{I}_n)$ is invertible.

In this subsection, we assume that $J_n = J_n^{(\mathrm{C})}$, the matrix introduced in subsection A-II.
For all $x \in \Lambda_n$, we set $s_n(x) = \sum_{k=1}^n x_i$. Verify that $$Z_n(h) = \mathrm{e}^{-\frac{\beta}{2}} \sum_{x \in \Lambda_n} \exp\left(\frac{\beta}{2n}\left(s_n(x)\right)^2 + h s_n(x)\right).$$

As in the third part, we assume that $B = A + \mathbf{u u}^T$ with $A \in \mathcal{S}_n(\mathbb{R})$ a symmetric matrix, and $\mathbf{u} \in \mathbb{R}^n$ a vector such that $\|\mathbf{u}\| = 1$. We denote by $\lambda_1 \leqslant \lambda_2 \leqslant \cdots \leqslant \lambda_n$ the eigenvalues of $A$ and $\mu_1 \leqslant \mu_2 \leqslant \cdots \leqslant \mu_n$ those of $B$. We admit that $$\lambda_1 \leqslant \mu_1 \leqslant \lambda_2 \leqslant \mu_2 \leqslant \cdots \leqslant \lambda_n \leqslant \mu_n.$$ We further assume that there exists an integer $m \in \{1,2,\ldots,n-1\}$ such that the eigenvalues of $A$ satisfy $$0 = \lambda_1 = \lambda_2 = \cdots = \lambda_m < \lambda_{m+1} \leqslant \cdots \leqslant \lambda_n.$$ Let $\varepsilon \in ]0, \lambda_{m+1}[$. We suppose that $\left\langle \mathbf{u}, \left(A - \varepsilon \mathbb{I}_n\right)^{-1} \mathbf{u} \right\rangle < -1$. Show that $(B - \varepsilon \mathbb{I}_n)$ is invertible.

As in the third part, we suppose that $B = A + \mathbf{u}\mathbf{u}^T$ with $A \in \mathcal{S}_n(\mathbb{R})$ a symmetric matrix, and $\mathbf{u} \in \mathbb{R}^n$ a vector such that $\|\mathbf{u}\| = 1$. We denote by $\lambda_1 \leqslant \lambda_2 \leqslant \cdots \leqslant \lambda_n$ the eigenvalues of $A$ and $\mu_1 \leqslant \mu_2 \leqslant \cdots \leqslant \mu_n$ those of $B$. We admit that $$\lambda_1 \leqslant \mu_1 \leqslant \lambda_2 \leqslant \mu_2 \leqslant \cdots \leqslant \lambda_n \leqslant \mu_n.$$ We further suppose that there exists an integer $m \in \{1, 2, \ldots, n-1\}$ such that the eigenvalues of $A$ satisfy $$0 = \lambda_1 = \lambda_2 = \cdots = \lambda_m < \lambda_{m+1} \leqslant \cdots \leqslant \lambda_n.$$ Let $\varepsilon \in ]0, \lambda_{m+1}[$. We suppose that $\left\langle \mathbf{u}, \left(A - \varepsilon \mathbb{I}_n\right)^{-1}\mathbf{u}\right\rangle < -1$.
Show that $\operatorname{Tr}\left(\left(B - \varepsilon \mathbb{I}_n\right)^{-1}\right) > \operatorname{Tr}\left(\left(A - \varepsilon \mathbb{I}_n\right)^{-1}\right)$.

As in the third part, we assume that $B = A + \mathbf{u u}^T$ with $A \in \mathcal{S}_n(\mathbb{R})$ a symmetric matrix, and $\mathbf{u} \in \mathbb{R}^n$ a vector such that $\|\mathbf{u}\| = 1$. We denote by $\lambda_1 \leqslant \lambda_2 \leqslant \cdots \leqslant \lambda_n$ the eigenvalues of $A$ and $\mu_1 \leqslant \mu_2 \leqslant \cdots \leqslant \mu_n$ those of $B$. We admit that $$\lambda_1 \leqslant \mu_1 \leqslant \lambda_2 \leqslant \mu_2 \leqslant \cdots \leqslant \lambda_n \leqslant \mu_n.$$ We further assume that there exists an integer $m \in \{1,2,\ldots,n-1\}$ such that the eigenvalues of $A$ satisfy $$0 = \lambda_1 = \lambda_2 = \cdots = \lambda_m < \lambda_{m+1} \leqslant \cdots \leqslant \lambda_n.$$ Let $\varepsilon \in ]0, \lambda_{m+1}[$. We suppose that $\left\langle \mathbf{u}, \left(A - \varepsilon \mathbb{I}_n\right)^{-1} \mathbf{u} \right\rangle < -1$. Show that $\operatorname{Tr}\left(\left(B - \varepsilon \mathbb{I}_n\right)^{-1}\right) > \operatorname{Tr}\left(\left(A - \varepsilon \mathbb{I}_n\right)^{-1}\right)$.

We fix an integer $d \in \mathbb{N}^*$, and we equip $\mathbb{R}^d$ with the usual inner product denoted $\langle \cdot, \cdot \rangle$ and the associated Euclidean norm $\|\cdot\|$. We denote $C := \{x \in \mathbb{R}^d \mid \|x\| \leq 1\}$ the closed unit ball of $\mathbb{R}^d$. Let $M$ be a nonzero real symmetric matrix of size $d \times d$ such that $\forall x \in \mathbb{R}^d,\, \langle x, Mx \rangle \geq 0$. We define $f(x) := -\frac{1}{2}\langle x, Mx \rangle$. Show that $\nabla f(x) = -Mx$, for all $x \in \mathbb{R}^d$.

We fix an integer $d \in \mathbb{N}^*$, and we equip $\mathbb{R}^d$ with the usual inner product denoted $\langle \cdot, \cdot \rangle$ and the associated Euclidean norm $\|\cdot\|$. We denote $C := \{x \in \mathbb{R}^d \mid \|x\| \leq 1\}$ the closed unit ball of $\mathbb{R}^d$. Let $M$ be a nonzero real symmetric matrix of size $d \times d$ such that $\forall x \in \mathbb{R}^d,\, \langle x, Mx \rangle \geq 0$. We define $f(x) := -\frac{1}{2}\langle x, Mx \rangle$. Describe the set of minimizers of $f$ on $C$.

A ``basic row operation'' on a matrix means adding a multiple of one row to another row. Consider the matrices $$A = \left(\begin{array}{rrr} x & 5 & x \\ 1 & 3 & -2 \\ -2 & -2 & 2 \end{array}\right) \quad \text{and} \quad B = \left(\begin{array}{rrr} 0 & 0 & 21 \\ 1 & -1 & -14 \\ 0 & \frac{4}{3} & 4 \end{array}\right)$$ It is given that $B$ can be obtained from $A$ by applying finitely many basic row operations. Then, the value of $x$ is:
(A) 3
(B) $-3$
(C) $-1$
(D) 2.

Let $\theta = \frac{2\pi}{7}$ and consider the following matrix $$A = \left(\begin{array}{rr} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{array}\right)$$ If $A^n$ means $A \times \cdots \times A$ ($n$ times), then $A^{100}$ is
(A) $\left(\begin{array}{rr} \cos 2\theta & -\sin 2\theta \\ \sin 2\theta & \cos 2\theta \end{array}\right)$
(B) $\left(\begin{array}{rr} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{array}\right)$
(C) $\left(\begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array}\right)$
(D) $\left(\begin{array}{rr} 0 & -1 \\ 1 & 0 \end{array}\right)$.

Let $A$ be a square matrix of real numbers such that $A ^ { 4 } = A$. Which of the following is true for every such $A$?
(A) $\operatorname { det } ( A ) \neq - 1$
(B) $A$ must be invertible.
(C) $A$ can not be invertible.
(D) $A ^ { 2 } + A + I = 0$ where $I$ denotes the identity matrix.

Let $n \geq 3$. Let $A = \left( \left( a _ { i j } \right) \right) _ { 1 \leq i , j \leq n }$ be an $n \times n$ matrix such that $a _ { i j } \in \{ 1 , - 1 \}$ for all $1 \leq i , j \leq n$. Suppose that $$\begin{aligned} & a _ { k 1 } = 1 \text { for all } 1 \leq k \leq n \text { and } \\ & \sum _ { k = 1 } ^ { n } a _ { k i } a _ { k j } = 0 \text { for all } i \neq j \end{aligned}$$ Show that $n$ is a multiple of 4.

If $A = \left( \begin{array} { l l } 1 & 1 \\ 0 & i \end{array} \right)$ and $A ^ { 2018 } = \left( \begin{array} { l l } a & b \\ c & d \end{array} \right)$, then $a + d$ equals:
(A) $1 + i$
(B) 0
(C) 2
(D) 2018.

Let $M$ be a $3 \times 3$ matrix with all entries being 0 or 1 . Then, all possible values for $\operatorname { det } ( M )$ are
(A) $0 , \pm 1$
(B) $0 , \pm 1 , \pm 2$
(C) $0 , \pm 1 , \pm 3$
(D) $0 , \pm 1 , \pm 2 , \pm 3$.

Consider all $2 \times 2$ matrices whose entries are distinct and taken from the set $\{ 1,2,3,4 \}$. The sum of determinants of all such matrices is
(A) 24 .
(B) 10 .
(C) 12 .
(D) 0 .

Define a polynomial $f ( x )$ by $$f ( x ) = \left| \begin{array} { l l l } 1 & x & x \\ x & 1 & x \\ x & x & 1 \end{array} \right|$$ for all $x \in \mathbb { R }$, where the right hand side above is a determinant. Then the roots of $f ( x )$ are of the form
(A) $\alpha , \beta \pm i \gamma$ where $\alpha , \beta , \gamma \in \mathbb { R } , \gamma \neq 0$ and $i$ is a square root of $- 1$.
(B) $\alpha , \alpha , \beta$ where $\alpha , \beta \in \mathbb { R }$ are distinct.
(C) $\alpha , \beta , \gamma$ where $\alpha , \beta , \gamma \in \mathbb { R }$ are all distinct.
(D) $\alpha , \alpha , \alpha$ for some $\alpha \in \mathbb { R }$.