Part A

We consider matrices $M$ of the form $M = \left( \begin{array} { l l } a & b \\ 5 & 3 \end{array} \right)$ where $a$ and $b$ are integers. The number $3 a - 5 b$ is called the determinant of $M$. We denote it $\operatorname { det } ( M )$. Thus $\operatorname { det } ( M ) = 3 a - 5 b$.

1. In this question we assume that $\operatorname { det } ( M ) \neq 0$ and we set $N = \frac { 1 } { \operatorname { det } ( M ) } \left( \begin{array} { c c } 3 & - b \\ - 5 & a \end{array} \right)$. Justify that $N$ is the inverse of $M$.
2. We consider the equation $( E ) : \quad \operatorname { det } ( M ) = 3$.
   We wish to determine all pairs of integers ( $a ; b$ ) that are solutions of equation ( $E$ ). a. Verify that the pair (6; 3) is a solution of $( E )$. b. Show that the pair of integers ( $a$; $b$ ) is a solution of ( $E$ ) if and only if $3 ( a - 6 ) = 5 ( b - 3 )$. Deduce the set of solutions of equation ( $E$ ).

Part B

1. We set $Q = \left( \begin{array} { l l } 6 & 3 \\ 5 & 3 \end{array} \right)$.
   Using Part A, determine the inverse matrix of $Q$.
2. Encoding with matrix $Q$
   To encode a two-letter word using the matrix $Q = \left( \begin{array} { l l } 6 & 3 \\ 5 & 3 \end{array} \right)$ we use the following procedure: Step 1: We associate with the word the matrix $X = \binom { x _ { 1 } } { x _ { 2 } }$ where $x _ { 1 }$ is the integer corresponding to the first letter of the word and $x _ { 2 }$ the integer corresponding to the second letter of the word according to the correspondence table below:
   

   A	B	C	D	E	F	G	H	I	J	K	L	M
   0	1	2	3	4	5	6	7	8	9	10	11	12
   N	O	P	Q	R	S	T	U	V	W	X	Y	Z
   13	14	15	16	17	18	19	20	21	22	23	24	25

   
   Step 2: The matrix $X$ is transformed into the matrix $Y = \binom { y _ { 1 } } { y _ { 2 } }$ such that $Y = Q X$. Step 3: The matrix $Y$ is transformed into the matrix $R = \binom { r _ { 1 } } { r _ { 2 } }$ such that $r _ { 1 }$ is the remainder of the Euclidean division of $y _ { 1 }$ by 26 and $r _ { 2 }$ is the remainder of the Euclidean division of $y _ { 2 }$ by 26.

A matriz $A = \begin{pmatrix} 2 & 1 \\ 3 & 4 \end{pmatrix}$ tem determinante igual a
(A) 2 (B) 3 (C) 5 (D) 8 (E) 11

QUESTION 156
The matrix $A = \begin{pmatrix} 2 & 1 \\ 3 & 4 \end{pmatrix}$ has determinant
(A) 2
(B) 3
(C) 4
(D) 5
(E) 6

The matrix $A = \begin{pmatrix} 2 & 1 \\ 3 & 4 \end{pmatrix}$ has determinant:
(A) 2
(B) 3
(C) 4
(D) 5
(E) 6

Let $m$ and $n$ be positive integers and $0 \leq k \leq \min\{m,n\}$ an integer. Prove or disprove: The subspace of $M_{m \times n}(\mathbb{C})$ consisting of all matrices of rank equal to $k$ is connected. (You may use the following fact: For $t \geq 2$, $\mathrm{GL}_{t}(\mathbb{C})$ is connected.)

For a $2 \times 2$ square matrix $X = \left( \begin{array} { l l } a & b \\ c & d \end{array} \right)$, $$D ( X ) = a d - b c$$ is defined. For a $2 \times 2$ square matrix $A = \left( \begin{array} { l l } 1 & 1 \\ 0 & p \end{array} \right)$, $$D \left( A ^ { 2 } \right) = D ( 5 A )$$ Find the sum of all constants $p$ that satisfy this condition. [4 points]

The system of linear equations in $x$ and $y$ $$\left( \begin{array} { c c } 5 - \log _ { 2 } a & 2 \\ 3 & \log _ { 2 } a \end{array} \right) \binom { x } { y } = \binom { 0 } { 0 }$$ has a solution other than $x = 0 , y = 0$. What is the sum of all values of $a$? [3 points]
(1) 8
(2) 10
(3) 12
(4) 16
(5) 20

For the system of linear equations in $x , y$:
$$\left( \begin{array} { l l } 5 & a \\ a & 3 \end{array} \right) \binom { x } { y } = \binom { x + 5 y } { 6 x + y }$$
Find the sum of all real values of $a$ such that the system has a solution other than $x = 0 , y = 0$. [4 points]

In the following graph, how many 1's are there among the components of the matrix representing the connection relationships between vertices? [3 points]
(1) 10
(2) 14
(3) 18
(4) 22
(5) 26

Given the determinant $\left| \begin{array} { l l l } 1 & a & c \\ 2 & d & b \\ 3 & 0 & 0 \end{array} \right| = 6$, find the determinant $\left| \begin{array} { l l } a & c \\ d & b \end{array} \right| =$ $\_\_\_\_$

Justify that, for every pair $(A , B)$ of elements of $\mathcal { M } ( n , \mathbb { K } )$, the matrix $[ A , B ]$ belongs to $\mathcal { M } _ { 0 } ( n , \mathbb { K } )$.

Let $A , B$ and $M$ be three elements of $\mathcal { M } _ { 0 } ( 2 , \mathbb { K } )$.
Express the trace of the matrix $M ^ { 2 }$ in terms of the determinant of $M$.

For $n \in \mathbb{N}^*$, $A \in \mathcal{S}_n(\mathbb{R})$ and $i \in \llbracket 1; n \rrbracket$, we denote by $A^{(i)}$ the square matrix of order $i$ extracted from $A$, consisting of the first $i$ rows and the first $i$ columns of $A$. For all $n \in \mathbb{N}^*$, we say that a matrix $A$ of $\mathcal{S}_n(\mathbb{R})$ satisfies property $\mathcal{P}_n$ if $\operatorname{det}\left(A^{(i)}\right) > 0$ for all $i \in \llbracket 1; n \rrbracket$.
In the special cases $n = 1$ and $n = 2$, show directly that any matrix $A \in \mathcal{S}_n(\mathbb{R})$ satisfying property $\mathcal{P}_n$ is positive definite.

For all $n \in \mathbb{N}^*$, we define the matrix $H_n$ by: $$\forall (i,j) \in \llbracket 1; n \rrbracket^2, \quad (H_n)_{i,j} = \frac{1}{i+j-1}$$ We also denote $\Delta_n = \operatorname{det}(H_n)$.
Calculate $H_2$ and $H_3$. Show that these are invertible matrices and determine their inverses.

For all $n \in \mathbb{N}^*$, we define the matrix $H_n$ by: $$\forall (i,j) \in \llbracket 1; n \rrbracket^2, \quad (H_n)_{i,j} = \frac{1}{i+j-1}$$ We also denote $\Delta_n = \operatorname{det}(H_n)$.
Show the relation: $$\Delta_{n+1} = \frac{(n!)^4}{(2n)!(2n+1)!} \Delta_n$$
Hint: you may start by subtracting the last column of $\Delta_{n+1}$ from all the others.

For all $n \in \mathbb{N}^*$, we define the matrix $H_n$ by: $$\forall (i,j) \in \llbracket 1; n \rrbracket^2, \quad (H_n)_{i,j} = \frac{1}{i+j-1}$$ We also denote $\Delta_n = \operatorname{det}(H_n)$.
Using the relation $\Delta_{n+1} = \frac{(n!)^4}{(2n)!(2n+1)!} \Delta_n$, deduce the expression of $\Delta_n$ as a function of $n$ (we will use the quantities $c_m = \prod_{i=1}^{m-1} i!$ for appropriate integers $m$).

Let $x$ be a linear recurrent sequence of minimal order $p \geqslant 1$ and minimal polynomial $B$. For any integer $n$ in $\mathbb{N}^*$, we denote by $H_n(x)$ the matrix in $\mathcal{M}_n(\mathbb{K})$ defined by $\forall (i,j) \in \{1,\ldots,n\}^2, [H_n(x)]_{i,j} = x_{i+j-2}$.
Show that the family $(\sigma^k(x))_{0 \leqslant k \leqslant p-1}$ is a basis of $\mathcal{R}_B(\mathbb{K})$.
Deduce from this, for any $n$ in $\mathbb{N}^*$, the rank of the family $(\sigma^k(x))_{0 \leqslant k \leqslant n-1}$.

Let $x$ be a linear recurrent sequence of minimal order $p \geqslant 1$ and minimal polynomial $B$. For any integer $n$ in $\mathbb{N}^*$, we denote by $H_n(x)$ the matrix in $\mathcal{M}_n(\mathbb{K})$ defined by $\forall (i,j) \in \{1,\ldots,n\}^2, [H_n(x)]_{i,j} = x_{i+j-2}$.
Show that if $n \geqslant p$, the map $\varphi_n : \left\{ \begin{array}{l} \mathcal{R}_B(\mathbb{K}) \rightarrow \mathbb{K}^n \\ v \mapsto (v_0, \ldots, v_{n-1}) \end{array} \right.$ is injective.
Deduce from this that if $n \geqslant p$, then $\operatorname{rang}(H_n(x)) = p$.
Remark: it is clear that this result remains true if $p = 0$ (since the sequence $x$ and the matrices $H_n(x)$ are zero).

Throughout this part $A$ and $B$ denote real symmetric matrices of $\mathcal { M } _ { 2 } ( \mathbb { R } )$. We set $$A = \left( \begin{array} { l l } a & b \\ b & d \end{array} \right)$$ and assume $A \geqslant 0$ (i.e., all eigenvalues of $A$ are $\geqslant 0$).
II.D.1) Prove that $\operatorname { det } ( A ) \geqslant 0$.
II.D.2) Prove that ${ } ^ { t } X A X \geqslant 0$ for every vector $X$.
II.D.3) Prove that $a \geqslant 0$ and $d \geqslant 0$.
II.D.4) Let $S \in \mathcal { M } _ { 2 } ( \mathbb { R } )$ be symmetric. Prove that: $$S \geqslant 0 \quad \text { if and only if } \quad ( \operatorname { Tr } ( S ) \geqslant 0 \text { and } \operatorname { det } ( S ) \geqslant 0 )$$

Throughout this part $A$ and $B$ denote real symmetric matrices of $\mathcal { M } _ { 2 } ( \mathbb { R } )$. We set $$A = \left( \begin{array} { l l } a _ { 1 } & b _ { 1 } \\ b _ { 1 } & d _ { 1 } \end{array} \right) \quad B = \left( \begin{array} { l l } a _ { 2 } & b _ { 2 } \\ b _ { 2 } & d _ { 2 } \end{array} \right)$$ We assume $A \geqslant 0$ and $B \geqslant 0$, $\operatorname { det } A \operatorname { det } B \neq 0$ and $b _ { 1 } b _ { 2 } \neq 0$.
II.F.1) Prove that we have equality in the formula of question II.E.2 if and only if the vectors $( a _ { 1 } , d _ { 1 } )$ and $( a _ { 2 } , d _ { 2 } )$ are linearly dependent, as well as the vectors $( b _ { 1 } , \sqrt { \operatorname { det } A }$ ) and $( b _ { 2 } , \sqrt { \operatorname { det } B }$ ).
II.F.2) Prove then that we have equality in the formula of question II.E.2 if and only if the matrices $A$ and $B$ are proportional ($A = \lambda B$ for some $\lambda \in \mathbb { R }$, $\lambda > 0$).

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: $\operatorname { det } ( A + B ) \geqslant \operatorname { det } A + \operatorname { det } B$.

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 $\det(E(A)) = e^{\operatorname{tr}(A)}$.