In this part all matrices are of format $(n, n)$, where $n$ is an integer greater than or equal to 2. We say that a real symmetric matrix is positive definite if and only if all its eigenvalues are strictly positive.
Let $A$ and $B$ be two positive definite symmetric matrices.
III.C.1) Prove that: $\operatorname { det } \left( I _ { n } + B \right) \geqslant 1 + \operatorname { det } B$.
III.C.2) Deduce that: $\operatorname { det } ( A + B ) \geqslant \operatorname { det } A + \operatorname { det } B$.

In this part all matrices are of format $(n, n)$, where $n$ is an integer greater than or equal to 2. We say that a real symmetric matrix is positive definite if and only if all its eigenvalues are strictly positive.
Let $A$ and $B$ be two positive definite symmetric matrices, $\alpha$ and $\beta$ two real numbers $> 0$ such that $\alpha + \beta = 1$; prove that: $$\operatorname { det } ( \alpha A + \beta B ) \geqslant ( \operatorname { det } A ) ^ { \alpha } ( \operatorname { det } B ) ^ { \beta }$$

In this part all matrices are of format $(n, n)$, where $n$ is an integer greater than or equal to 2. We say that a real symmetric matrix is positive definite if and only if all its eigenvalues are strictly positive.
For $1 \leqslant i \leqslant k$, let $A _ { i }$ be positive definite symmetric matrices and $\alpha _ { i }$ strictly positive real numbers such that $\alpha _ { 1 } + \cdots + \alpha _ { k } = 1$. Prove that $$\operatorname { det } \left( \alpha _ { 1 } A _ { 1 } + \cdots + \alpha _ { k } A _ { k } \right) \geqslant \left( \operatorname { det } A _ { 1 } \right) ^ { \alpha _ { 1 } } \ldots \left( \operatorname { det } A _ { k } \right) ^ { \alpha _ { k } }$$
One may reason by induction on $k$.

Show that $\mathrm{GL}_n(\mathbb{R})$ is a dense subset of $\mathcal{M}_n(\mathbb{R})$.

For $p \in \mathbb{N}^*$, we set $$A_p = \left(\begin{array}{rrrrr} 2 & -1 & 0 & \cdots & 0 \\ -1 & 2 & -1 & \ddots & \vdots \\ 0 & -1 & 2 & \ddots & 0 \\ \vdots & \ddots & \ddots & \ddots & -1 \\ 0 & \cdots & 0 & -1 & 2 \end{array}\right) \in \mathcal{M}_p(\mathbb{R})$$ We denote by $P_p$ the polynomial such that, for all real $x$, $P_p(x) = \det(x I_p - A_p)$.
Show that for fixed $x \in \mathbb{R}$, the sequence $(P_p(x))_{p \in \mathbb{N}^*}$ satisfies a linear relation of order 2, which we shall specify.

For $p \in \mathbb{N}^*$, we set $$A_p = \left(\begin{array}{rrrrr} 2 & -1 & 0 & \cdots & 0 \\ -1 & 2 & -1 & \ddots & \vdots \\ 0 & -1 & 2 & \ddots & 0 \\ \vdots & \ddots & \ddots & \ddots & -1 \\ 0 & \cdots & 0 & -1 & 2 \end{array}\right) \in \mathcal{M}_p(\mathbb{R})$$ We denote by $P_p$ the polynomial such that, for all real $x$, $P_p(x) = \det(x I_p - A_p)$.
Let $x \in \mathbb{R}$ such that $|2 - x| < 2$. After justifying the existence of a unique $\theta \in ]0, \pi[$ such that $2 - x = 2\cos\theta$, determine $P_p(x)$ as a function of $\sin((p+1)\theta)$ and $\sin(\theta)$.

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

We are given a continuous function $\xi : \mathbb{R} \rightarrow \mathbb{R}$ satisfying condition (V.1) (with $d \geqslant 2$), where $$\forall A \in \mathcal{M}_d(\mathbb{R}), \quad A \text{ invertible} \Rightarrow f_\xi(A) = \left(\xi(A_{i,j})\right)_{1\leqslant i,j\leqslant d} \text{ invertible} \tag{V.1}$$
Show $$\forall (a, b, c, d) \in \mathbb{R}^4, \quad ad \neq bc \Rightarrow \xi(a)\xi(d) \neq \xi(b)\xi(c)$$ One may consider the matrix $\begin{pmatrix} a & b & 0 & \cdots & 0 \\ c & d & 0 & \cdots & 0 \\ c & d & & & \\ \vdots & \vdots & & I_{d-2} & \\ c & d & & & \end{pmatrix}$.

For $\lambda \in \mathbb{R}$, calculate the determinant of the matrix $A_\lambda \in \mathcal{M}_d(\mathbb{R})$ having only 1's off the diagonal and only $\lambda$ on the diagonal.

For $\lambda \in \mathbb{R}$, let $A_\lambda \in \mathcal{M}_d(\mathbb{R})$ be the matrix having only 1's off the diagonal and only $\lambda$ on the diagonal.
Deduce all continuous functions $\xi : \mathbb{R} \rightarrow \mathbb{R}$ satisfying $$\forall A \in \mathcal{M}_d(\mathbb{R}), \quad A \text{ invertible} \Rightarrow f_\xi(A) = \left(\xi(A_{i,j})\right)_{1\leqslant i,j\leqslant d} \text{ invertible} \tag{V.1}$$

Let $f$ denote a function of class $C^1$ from $\mathbb{R}^n$ to $\mathbb{R}^n$ satisfying $f(0) = 0$. For $t$ real and $j$ an integer in $\llbracket 1, n \rrbracket$, we denote by $t_j$ the element $(0, \ldots, 0, t, 0, \ldots, 0)$ of $\mathbb{R}^n$, the real number $t$ being in position $j$.
We admit that if functions $\varphi_1, \varphi_2, \ldots, \varphi_n$ are continuous on $\mathbb{R}$ and take values in $\mathbb{R}^n$, then the function $\Phi$ defined on $\mathbb{R}$ by: $$\Phi(t) = \operatorname{det}(\varphi_1(t), \varphi_2(t), \ldots, \varphi_n(t))$$ is continuous on $\mathbb{R}$.
Using question I.B.2 and the multilinearity of the determinant, show that in a neighbourhood of 0 $$\operatorname{det}\left(f(t_1), f(t_2), \ldots, f(t_n)\right) = t^n \mathrm{jac}_f(0) + \mathrm{o}\left(t^n\right)$$

Let $f$ denote a function of class $C^1$ from $\mathbb{R}^n$ to $\mathbb{R}^n$ satisfying $f(0) = 0$. For $t$ real and $j$ an integer in $\llbracket 1, n \rrbracket$, we denote by $t_j$ the element $(0, \ldots, 0, t, 0, \ldots, 0)$ of $\mathbb{R}^n$, the real number $t$ being in position $j$.
Deduce that $$\lim_{t \to 0} \frac{\operatorname{det}\left(f(t_1), \ldots, f(t_n)\right)}{\operatorname{det}\left(t_1, \ldots, t_n\right)} = \mathrm{jac}_f(0)$$

Let $f$ denote a function of class $C^1$ from $\mathbb{R}^n$ to $\mathbb{R}^n$ satisfying $f(0) = 0$. For $t$ real and $j$ an integer in $\llbracket 1, n \rrbracket$, we denote by $t_j$ the element $(0, \ldots, 0, t, 0, \ldots, 0)$ of $\mathbb{R}^n$, the real number $t$ being in position $j$.
In the case $n = 2$ (respectively $n = 3$), give a geometric interpretation of the absolute value of the Jacobian of $f$ at 0 using areas of parallelograms (respectively volumes of parallelepipeds).

We denote by $\mathrm{GL}_2(\mathbb{Z})$ the set of invertible elements of the ring $\mathcal{M}_2(\mathbb{Z})$, equipped with its usual addition and multiplication.
Justify that an element $M$ of $\mathcal{M}_2(\mathbb{Z})$ belongs to $\mathrm{GL}_2(\mathbb{Z})$ if and only if $|\det M| = 1$.

Let $\mathcal { B } : = \left( e _ { 1 } , \ldots , e _ { n } \right)$ be a basis of $V$. We associate to every symmetric bilinear form $b$ on $V$ a symmetric matrix $\Phi _ { \mathcal { B } } ( b ) : = \left( b \left( e _ { i } , e _ { j } \right) \right) _ { i , j = 1 \ldots n }$ called the matrix of $b$ in the basis $\mathcal { B }$. We recall that $b \mapsto \Phi _ { \mathcal { B } } ( b )$ is an isomorphism between the vector space of symmetric bilinear forms on $V$ and that of square symmetric matrices of size $n$.
(a) Prove that a quadratic form $q$ on $V$ is non-degenerate if and only if the determinant $\operatorname { det } \left( \Phi _ { \mathcal { B } } ( \tilde { q } ) \right)$ is non-zero.
(b) What is the matrix of $\left\langle a _ { 1 } , \ldots , a _ { n } \right\rangle$ in the canonical basis of $\mathbb { K } ^ { n }$ ? Deduce that $\left\langle a _ { 1 } , \ldots , a _ { n } \right\rangle \in \mathcal { Q } \left( \mathbb { K } ^ { n } \right)$.

Let $\mathcal{U}, \mathcal{V}$ and $\mathcal{W}$ be three vector subspaces of $\mathbb{R}^{n}$ such that $$\operatorname{dim} \mathcal{U} + \operatorname{dim} \mathcal{V} + \operatorname{dim} \mathcal{W} > 2n.$$ Show that $\mathcal{U} \cap \mathcal{V} \cap \mathcal{W}$ is not reduced to $\{0\}$.

Let $r$ and $s$ be two non-zero natural integers. Let $A \in \mathcal { M } _ { r } ( \mathbb { R } ) , B \in \mathcal { M } _ { r , s } ( \mathbb { R } ) , C \in \mathcal { M } _ { s , r } ( \mathbb { R } )$ and $D \in \mathcal { M } _ { s } ( \mathbb { R } )$. We further assume that $A$ is invertible. We consider the matrix $M \in \mathcal { M } _ { r + s } ( \mathbb { R } )$ having the following block form $$M = \left[ \begin{array} { l l } A & B \\ C & D \end{array} \right]$$ Find two matrices $U \in \mathcal { M } _ { r , s } ( \mathbb { R } )$ and $V \in \mathcal { M } _ { s } ( \mathbb { R } )$ such that $$M = \left[ \begin{array} { c c } A & 0 \\ C & I _ { s } \end{array} \right] \cdot \left[ \begin{array} { c c } I _ { r } & U \\ 0 & V \end{array} \right]$$ and deduce that $$\operatorname { det } ( M ) = \operatorname { det } ( A ) \cdot \operatorname { det } \left( D - C A ^ { - 1 } B \right)$$

Prove that for all $M \in \mathcal{Y}_n$, $\operatorname{det}(M) \leqslant n!$ and that there is no equality.

Justify that the determinant has a maximum on $\mathcal{X}_n$ (denoted $x_n$) and a maximum on $\mathcal{Y}_n$ (denoted $y_n$).

Let $J \in \mathcal{X}_n$ be the matrix whose coefficients all equal 1. We set $M = J - I_n$.
Calculate $\operatorname{det}(M)$ and deduce that $\lim_{k \to +\infty} y_k = +\infty$.

Let $N = (n_{i,j})_{i,j} \in \mathcal{Y}_n$. Fix $1 \leqslant i, j \leqslant n$ and suppose that $n_{i,j} \in ]0,1[$.
Prove that by replacing $n_{i,j}$ either by 0 or by 1, we can obtain a matrix $N'$ of $\mathcal{Y}_n$ such that $\operatorname{det}(N) \leqslant \operatorname{det}(N')$.
Deduce that $x_n = y_n$.

Prove that if $M \in \mathrm{O}_n(\mathbb{R})$, then its determinant equals 1 or $-1$. What do you think of the converse?

Let $M \in M_n(\mathbb{R})$ be an invertible matrix with integer coefficients.
1a. Show that $M^{-1}$ has rational coefficients.
1b. Show the equivalence of the following propositions:
i) $M^{-1}$ has integer coefficients.
ii) $\det M$ equals 1 or $-1$.

We consider $A \in \mathcal{M}_{n}(\mathbb{R})$. We assume $A$ is invertible and, in accordance with the notations of the problem, $(A^{-1})_{s}$ denotes the symmetric part of the inverse of $A$. Show that $(\operatorname{det}(A))^{2}\operatorname{det}\left((A^{-1})_{s}\right) = \operatorname{det}(A_{s})$.
One may consider $A(A^{-1})_{s}A^{\top}$.

We denote $E_{n} = \mathcal{M}_{n,1}(\mathbb{R})$ equipped with the inner product $(X \mid Y) = X^{\top}Y$. A matrix $K \in \mathcal{M}_{n}(\mathbb{R})$ is called $F$-singular if there exists a non-zero $X \in F$ such that $\forall Z \in F, Z^{\top}KX = 0$. We assume $n \geqslant 2$. Let $F = H$ be a hyperplane of $E_{n}$ and let $N \in E_{n}$ be a unit vector normal to $H$.
Deduce that $A$ is $H$-singular if and only if the matrix $A_{N} = \begin{pmatrix} A & N \\ N^{\top} & 0 \end{pmatrix} \in \mathcal{M}_{n+1}(\mathbb{R})$ is singular.