In this question we consider the linear differential system $\mathcal{S} : X' = AX$ associated with the matrix $A = \begin{pmatrix} 1 & -4 & 0 \\ 1 & -2 & -1 \\ 1 & 1 & 0 \end{pmatrix}$.
We call trajectories of $\mathcal{S}$ the arcs of the space $\mathbb{R}^3$ parametrized by the solutions of $\mathcal{S}$. We want to determine the rectilinear trajectories and the planar trajectories of $\mathcal{S}$.
IV.F.1) Construct an invertible matrix $P$ and a matrix $T = \begin{pmatrix} \alpha & \beta & 0 \\ -\beta & \alpha & 0 \\ 0 & 0 & \gamma \end{pmatrix}$ with $(\alpha, \beta, \gamma)$ in $(\mathbb{R}^*)^3$ such that $P^{-1}AP = T$ and determine a plane $F$ and a line $G$ stable by the endomorphism of $\mathbb{R}^3$ canonically associated with $A$ and supplementary in $\mathbb{R}^3$.
IV.F.2) Determine the unique solution of the Cauchy problem $\mathcal{P}_U : \begin{cases} X' = AX \\ X(0) = U \end{cases}$ when $U$ belongs to $G$.
IV.F.3) For all $\sigma = (a, b)$ in $\mathbb{R}^2$, we consider the Cauchy problem $\mathcal{C}_{\sigma} : \begin{cases} x' = -x + 2y \\ y' = -2x - y \\ x(0) = a,\ y(0) = b \end{cases}$ and $\varphi = (x, y)$ in $\mathcal{C}^1(\mathbb{R}, \mathbb{R}^2)$ the unique solution of $\mathcal{C}_{\sigma}$.
Specify $x'(0)$ and $y'(0)$; show that $x$ and $y$ are solutions of the same linear homogeneous differential equation of second order with constant coefficients and thus deduce $\varphi$ as a function of $a$ and $b$.
IV.F.4) Determine the rectilinear trajectories and the planar trajectories of the differential system $X' = AX$.

Let $A$ be a continuous function, periodic of period $T > 0$ and $X$ a function of class $\mathcal { C } ^ { 1 }$ $$A : \left\lvert \, \begin{aligned} & \mathbb { R } \rightarrow \mathcal { M } _ { 2 } ( \mathbb { C } ) \\ & t \mapsto A ( t ) \end{aligned} \quad X \right. : \begin{aligned} & \mathbb { R } \rightarrow \mathbb { C } ^ { 2 } \\ & t \mapsto \binom { x _ { 1 } ( t ) } { x _ { 2 } ( t ) } \end{aligned}$$ We are interested in the homogeneous differential system with unknown $X$ $$\forall t \in \mathbb { R } , \quad X ^ { \prime } ( t ) = A ( t ) X ( t ) \tag{IV.1}$$ We fix $t _ { 0 } \in \mathbb { R }$. We denote by $U$ and $V$ the two solutions of the differential system (IV.1) satisfying $U \left( t _ { 0 } \right) = \binom { 1 } { 0 }$ and $V \left( t _ { 0 } \right) = \binom { 0 } { 1 }$.
We consider the linear differential system (IV.2) whose solutions are functions of class $\mathcal { C } ^ { 1 }$ with values in $\mathcal { M } _ { 2 } ( \mathbb { C } )$ $$\forall t \in \mathbb { R } , \quad M ^ { \prime } ( t ) = A ( t ) M ( t ) \tag{IV.2}$$ For all $t \in \mathbb { R }$, we set $E ( t ) = [ U ( t ) , V ( t ) ]$. Verify that $E$ is the solution of (IV.2) satisfying $E \left( t _ { 0 } \right) = I _ { 2 }$.

Let $A$ be a continuous function, periodic of period $T > 0$. We are interested in the homogeneous differential system $$\forall t \in \mathbb { R } , \quad X ^ { \prime } ( t ) = A ( t ) X ( t ) \tag{IV.1}$$ We consider the linear differential system $$\forall t \in \mathbb { R } , \quad M ^ { \prime } ( t ) = A ( t ) M ( t ) \tag{IV.2}$$ If $M : \left\lvert \, \begin{aligned} & \mathbb { R } \rightarrow \mathcal { M } _ { 2 } ( \mathbb { C } ) \\ & t \mapsto [ F ( t ) , G ( t ) ] \end{aligned} \right.$ is a solution of (IV.2) and $W = \binom { w _ { 1 } } { w _ { 2 } } \in \mathbb { C } ^ { 2 }$, prove that the function $Y : \begin{aligned} & \mathbb { R } \rightarrow \mathbb { C } ^ { 2 } \\ & t \mapsto M ( t ) W = w _ { 1 } F ( t ) + w _ { 2 } G ( t ) \end{aligned}$ is a solution of (IV.1).

Let $A$ be a continuous function, periodic of period $T > 0$. We are interested in the homogeneous differential system $$\forall t \in \mathbb { R } , \quad X ^ { \prime } ( t ) = A ( t ) X ( t ) \tag{IV.1}$$ We fix $t _ { 0 } \in \mathbb { R }$. We denote by $U$ and $V$ the two solutions of (IV.1) satisfying $U \left( t _ { 0 } \right) = \binom { 1 } { 0 }$ and $V \left( t _ { 0 } \right) = \binom { 0 } { 1 }$, and set $E ( t ) = [ U ( t ) , V ( t ) ]$.
Let $t _ { 1 } \in \mathbb { R }$ and $W = \binom { w _ { 1 } } { w _ { 2 } } \in \mathbb { C } ^ { 2 }$. Assume that $E \left( t _ { 1 } \right) W = \binom { 0 } { 0 }$. Show that the function $Y : \left\lvert \, \begin{aligned} & \mathbb { R } \rightarrow \mathbb { C } ^ { 2 } \\ & t \mapsto E ( t ) W = w _ { 1 } U ( t ) + w _ { 2 } V ( t ) \end{aligned} \right.$ is zero. Deduce that for all real $t , E ( t )$ is invertible.

Let $A$ be a continuous function, periodic of period $T > 0$. We are interested in the homogeneous differential system $$\forall t \in \mathbb { R } , \quad X ^ { \prime } ( t ) = A ( t ) X ( t ) \tag{IV.1}$$ We fix $t _ { 0 } \in \mathbb { R }$ and set $E ( t ) = [ U ( t ) , V ( t ) ]$ where $U, V$ are the solutions of (IV.1) with $U(t_0) = \binom{1}{0}$, $V(t_0) = \binom{0}{1}$. We consider the linear differential system $$\forall t \in \mathbb { R } , \quad M ^ { \prime } ( t ) = A ( t ) M ( t ) \tag{IV.2}$$ Let $M \in \mathcal { C } ^ { 1 } \left( \mathbb { R } , \mathcal { M } _ { 2 } ( \mathbb { C } ) \right)$ be a solution of system (IV.2). Show that for all real $t , M ( t ) = E ( t ) M \left( t _ { 0 } \right)$.

Let $A$ be a continuous function, periodic of period $T > 0$. We are interested in the homogeneous differential system $$\forall t \in \mathbb { R } , \quad X ^ { \prime } ( t ) = A ( t ) X ( t ) \tag{IV.1}$$ We fix $t _ { 0 } \in \mathbb { R }$ and set $E ( t ) = [ U ( t ) , V ( t ) ]$ where $U, V$ are the solutions of (IV.1) with $U(t_0) = \binom{1}{0}$, $V(t_0) = \binom{0}{1}$. Deduce from the previous question that there exists a unique matrix $B \in \mathrm { GL } _ { 2 } ( \mathbb { C } )$ independent of $t$ such that for all real $t , E ( t + T ) = E ( t ) B$.

Let $A$ be a continuous function, periodic of period $T > 0$. We are interested in the homogeneous differential system $$\forall t \in \mathbb { R } , \quad X ^ { \prime } ( t ) = A ( t ) X ( t ) \tag{IV.1}$$ We fix $t _ { 0 } \in \mathbb { R }$ and set $E ( t ) = [ U ( t ) , V ( t ) ]$ where $U, V$ are the solutions of (IV.1) with $U(t_0) = \binom{1}{0}$, $V(t_0) = \binom{0}{1}$. $B \in \mathrm { GL } _ { 2 } ( \mathbb { C } )$ is the unique matrix such that $E(t+T) = E(t)B$ for all $t$. The Floquet multipliers of (IV.1) are the eigenvalues of $B$.
Let $\rho \in \mathbb { C }$ be a Floquet multiplier of (IV.1) and $Z \in \mathbb { C } ^ { 2 }$ be an eigenvector of $B$ associated with this eigenvalue. We denote $Y : \begin{aligned} & \mathbb { R } \rightarrow \mathbb { C } ^ { 2 } \\ & t \mapsto E ( t ) Z \end{aligned}$.
a) Prove that $\forall t \in \mathbb { R } , Y ( t + T ) = \rho Y ( t )$.
b) Prove that there exists a complex number $\mu$ and a function $S : \left\lvert \, \begin{aligned} & \mathbb { R } \rightarrow \mathbb { C } ^ { 2 } \\ & t \mapsto S ( t ) \end{aligned} \right.$ non-zero and $T$-periodic such that $\forall t \in \mathbb { R } , Y ( t ) = \mathrm { e } ^ { \mu t } S ( t )$.

Let $A$ be a continuous function, periodic of period $T > 0$. We are interested in the homogeneous differential system $$\forall t \in \mathbb { R } , \quad X ^ { \prime } ( t ) = A ( t ) X ( t ) \tag{IV.1}$$ $B \in \mathrm { GL } _ { 2 } ( \mathbb { C } )$ is the Floquet matrix of (IV.1) and its eigenvalues are the Floquet multipliers. Give a necessary and sufficient condition on the Floquet multipliers for the differential system (IV.1) to admit a non-zero periodic solution of period $T$.

Let $A$ be a continuous function, periodic of period $T > 0$. We are interested in the homogeneous differential system $$\forall t \in \mathbb { R } , \quad X ^ { \prime } ( t ) = A ( t ) X ( t ) \tag{IV.1}$$ $B \in \mathrm { GL } _ { 2 } ( \mathbb { C } )$ is the Floquet matrix of (IV.1) and its eigenvalues are the Floquet multipliers. Assume that the matrix $B$ is diagonalizable. Give a necessary and sufficient condition on the Floquet multipliers for the differential system (IV.1) to admit an unbounded solution on $\mathbb { R }$.

Let $A$ be a continuous function, periodic of period $T > 0$. We are interested in the homogeneous differential system $$\forall t \in \mathbb { R } , \quad X ^ { \prime } ( t ) = A ( t ) X ( t ) \tag{IV.1}$$ We fix $t _ { 0 } \in \mathbb { R }$ and set $E ( t ) = [ U ( t ) , V ( t ) ]$ where $U, V$ are the solutions of (IV.1) with $U(t_0) = \binom{1}{0}$, $V(t_0) = \binom{0}{1}$. We set for all $t \in \mathbb { R } , W ( t ) = \operatorname { det } ( E ( t ) )$. Show that for all real $t , W ^ { \prime } ( t ) = \operatorname { tr } ( A ( t ) ) W ( t )$.

Let $A$ be a continuous function, periodic of period $T > 0$. We are interested in the homogeneous differential system $$\forall t \in \mathbb { R } , \quad X ^ { \prime } ( t ) = A ( t ) X ( t ) \tag{IV.1}$$ We fix $t _ { 0 } \in \mathbb { R }$ and set $E ( t ) = [ U ( t ) , V ( t ) ]$ where $U, V$ are the solutions of (IV.1) with $U(t_0) = \binom{1}{0}$, $V(t_0) = \binom{0}{1}$. We set $W ( t ) = \operatorname { det } ( E ( t ) )$ and denote $\rho _ { 1 }$ and $\rho _ { 2 }$ the Floquet multipliers of (IV.1). Deduce that $\rho _ { 1 } \rho _ { 2 } = \exp \left( \int _ { 0 } ^ { T } \operatorname { tr } ( A ( s ) ) \mathrm { d } s \right)$.

Let $M \in M_{n}(\mathbf{R})$. We denote by (S) the differential system: $$(\mathrm{S}) \quad X' = MX$$ where $X$ is a function from the variable $t$ in $\mathbf{R}$ to $\mathbf{R}^{n}$, differentiable on $\mathbf{R}$.
Let $T \in M_{n}(\mathbf{R})$. We assume that $M$ is similar to $T$ in $M_{n}(\mathbf{R})$ and we denote by $(\mathrm{S}^{*})$ the differential system $$(\mathrm{S}^{*}) \quad Y' = TY$$
Prove that the coordinates of a solution $X$ of (S) are linear combinations of the coordinates of a solution $Y$ of $(\mathrm{S}^{*})$.

We consider two distinct complex numbers $\alpha$ and $\beta$. We assume that a matrix $A \in \mathcal{M}_3(\mathbf{C})$ has $\alpha$ as a simple eigenvalue and $\beta$ as a double eigenvalue.
$\mathbf{18}$ ▷ Show that $A$ is similar to a matrix of the form $$T = \left( \begin{array}{ccc} \alpha & 0 & 0 \\ 0 & \beta & a \\ 0 & 0 & \beta \end{array} \right)$$ where $a$ is a certain complex number. Compute $T^n$ for $n$ a natural integer, then $e^{tT}$ for $t$ real. Deduce from this a necessary and sufficient condition on $\alpha$ and $\beta$ for $\lim_{t \rightarrow +\infty} e^{tA} = 0_3$.

Let $M \in M_{n}(\mathbf{R})$. We denote by (S) the differential system: $$(\mathrm{S}) \quad X' = MX$$ where $X$ is a function from the variable $t$ in $\mathbf{R}$ to $\mathbf{R}^{n}$, differentiable on $\mathbf{R}$.
We assume $n = 2$, we then denote $X = (x; y)$ where $x$ and $y$ are two functions differentiable from $\mathbf{R}$ to $\mathbf{R}$ and we set $z = x + iy$.
We assume that there exist real numbers $a$ and $b$ such that $M = \left(\begin{array}{cc} a & b \\ -b & a \end{array}\right)$.
Prove that $X$ is a solution of (S) if and only if $z$ is a solution of a first-order linear differential equation to be determined. Deduce an expression, as a function of $t$, of the coordinates of the solutions of (S).
Solve the system $X' = BX$ where $B$ is the matrix from question 2).

Let $M \in M_{n}(\mathbf{R})$. We denote by (S) the differential system: $$(\mathrm{S}) \quad X' = MX$$ where $X$ is a function from the variable $t$ in $\mathbf{R}$ to $\mathbf{R}^{n}$, differentiable on $\mathbf{R}$.
Let $T \in M_{n}(\mathbf{R})$. We assume that $M$ is similar to $T$ in $M_{n}(\mathbf{R})$ and we denote by $(\mathrm{S}^{*})$ the differential system $$(\mathrm{S}^{*}) \quad Y' = TY$$
We consider the following assertions:

• [$\mathbf{A}_{1}$] $\chi_{M}$ is a Hurwitz polynomial;
• [$\mathbf{A}_{2}$] The solutions of (S) tend to $0_{\mathbf{R}^{n}}$ as $t$ tends to $+\infty$;
• [$\mathbf{A}_{3}$] There exist $\alpha > 0$ and $k > 0$ such that for every solution $\Phi$ of (S), $$\forall t \geq 0 \quad : \quad \|\Phi(t)\| \leq k e^{-\alpha t} \|\Phi(0)\|.$$

Let $T \in M_{n}(\mathbf{R})$. We assume that $T$ satisfies the following condition: $$(\mathrm{C}) \quad \exists \beta \in \mathbf{R}_{+}^{*}, \forall X \in \mathbf{R}^{n} : \langle TX, X \rangle \leq -\beta \|X\|^{2}.$$
Prove that $\mathrm{A}_{3}$ is true with $k = 1$ for every solution $\Phi$ of $(\mathrm{S}^{*})$.
Hint: one may introduce the function $t \mapsto e^{2\beta t} \|\Phi(t)\|^{2}$.

Let $M \in M_{n}(\mathbf{R})$. We denote by (S) the differential system: $$(\mathrm{S}) \quad X' = MX$$ where $X$ is a function from the variable $t$ in $\mathbf{R}$ to $\mathbf{R}^{n}$, differentiable on $\mathbf{R}$.
We consider the following assertions:

• [$\mathbf{A}_{1}$] $\chi_{M}$ is a Hurwitz polynomial;
• [$\mathbf{A}_{2}$] The solutions of (S) tend to $0_{\mathbf{R}^{n}}$ as $t$ tends to $+\infty$;
• [$\mathbf{A}_{3}$] There exist $\alpha > 0$ and $k > 0$ such that for every solution $\Phi$ of (S), $$\forall t \geq 0 \quad : \quad \|\Phi(t)\| \leq k e^{-\alpha t} \|\Phi(0)\|.$$

Assume that $M \in M_{n}(\mathbf{R})$ is semi-simple. Prove that the assertions $\mathrm{A}_{1}$, $\mathrm{A}_{2}$ and $\mathrm{A}_{3}$ are equivalent.
Hint: one may start with $A_{3}$ implies $A_{2}$.

In this part, $a$ denotes an endomorphism of $\mathbf { C } ^ { n }$. We use the decomposition $\mathbf { C } ^ { n } = \bigoplus _ { i = 1 } ^ { r } E _ { i }$ where $E _ { i } = \operatorname { Ker } \left( a - \lambda _ { i } id _ { \mathbf { C } ^ { n } } \right) ^ { m _ { i } }$, with the projections $p_i$, inclusions $q_i$, $a_i = p_i a q_i$, and $a = \sum _ { i = 1 } ^ { r } q _ { i } a _ { i } p _ { i }$.
Deduce that: $$\forall t \in \mathbf { R } , \quad e ^ { t a } = \sum _ { i = 1 } ^ { r } q _ { i } e ^ { t a _ { i } } p _ { i }$$

In this part, $a$ denotes an endomorphism of $\mathbf { C } ^ { n }$. We use the decomposition $\mathbf { C } ^ { n } = \bigoplus _ { i = 1 } ^ { r } E _ { i }$ where $E _ { i } = \operatorname { Ker } \left( a - \lambda _ { i } id _ { \mathbf { C } ^ { n } } \right) ^ { m _ { i } }$, with $a_i = p_i a q_i$ the endomorphism of $E_i$ and $\|.\|_i$ the norm on $\mathcal{L}(E_i)$.
Show moreover that: $$\forall i \in \llbracket 1 ; r \rrbracket , \quad \forall t \in \mathbf { R } , \quad \left\| e ^ { t a _ { i } } \right\| _ { i } \leqslant \left| e ^ { t \lambda _ { i } } \right| \sum _ { k = 0 } ^ { m _ { i } - 1 } \frac { | t | ^ { k } } { k ! } \left\| a _ { i } - \lambda _ { i } id _ { E _ { i } } \right\| _ { i } ^ { k }$$

Let $T > 0$ be a real number and $n \in \mathbb{N}^*$ be a natural integer. Let $A : \mathbb{R} \rightarrow \mathscr{M}_n(\mathbb{C})$ be a continuous application on $\mathbb{R}$ and $T$-periodic. We consider the differential system $$X'(t) = A(t) X(t) \tag{2}$$ where $X$ is a function from $\mathbb{R}$ to $\mathbb{C}^n$, of class $\mathscr{C}^1$ on $\mathbb{R}$.
Show that there exists $\mu \in \mathbb{C}^*$ and a non-zero solution $Y \in \mathscr{C}^1(\mathbb{R}, \mathbb{C}^n)$ of (2) such that $$\forall t \in \mathbb{R}, \quad Y(t+T) = \mu Y(t)$$

Let $T > 0$ be a real number and $n \in \mathbb{N}^*$ be a natural integer. Let $A : \mathbb{R} \rightarrow \mathscr{M}_n(\mathbb{C})$ be an application continuous on $\mathbb{R}$ and $T$-periodic. We consider the differential system $$X'(t) = A(t) X(t) \tag{2}$$ where $X$ is a function from $\mathbb{R}$ to $\mathbb{C}^n$, of class $\mathscr{C}^1$ on $\mathbb{R}$.
Show that there exists $\mu \in \mathbb{C}^*$ and a non-zero solution $Y \in \mathscr{C}^1(\mathbb{R}, \mathbb{C}^n)$ of (2) such that $$\forall t \in \mathbb{R}, \quad Y(t+T) = \mu Y(t).$$

Let $T > 0$ be a real number and $n \in \mathbb{N}^*$ be a natural integer. Let $A : \mathbb{R} \rightarrow \mathscr{M}_n(\mathbb{C})$ be a continuous application on $\mathbb{R}$ and $T$-periodic. We consider the differential system $$X'(t) = A(t) X(t) \tag{2}$$ where $X$ is a function from $\mathbb{R}$ to $\mathbb{C}^n$, of class $\mathscr{C}^1$ on $\mathbb{R}$. Let $\mathscr{S}$ be the space of solutions in $\mathscr{C}^1(\mathbb{R}, \mathbb{C}^n)$ of (2). Let $(Y_1, Y_2, \ldots, Y_n)$ be a basis of $\mathscr{S}$. For $t \in \mathbb{R}$, we denote by $M(t)$ the matrix whose columns are $Y_1(t), \ldots, Y_n(t)$.
Show that for every real number $t$, $M(t) \in \mathrm{GL}_n(\mathbb{C})$ and $M'(t) = A(t) M(t)$.

Let $T > 0$ be a real number and $n \in \mathbb{N}^*$ be a natural integer. Let $A : \mathbb{R} \rightarrow \mathscr{M}_n(\mathbb{C})$ be a continuous application on $\mathbb{R}$ and $T$-periodic. We consider the differential system $$X'(t) = A(t) X(t) \tag{2}$$ where $X$ is a function from $\mathbb{R}$ to $\mathbb{C}^n$, of class $\mathscr{C}^1$ on $\mathbb{R}$. Let $\mathscr{S}$ be the space of solutions in $\mathscr{C}^1(\mathbb{R}, \mathbb{C}^n)$ of (2). Let $(Y_1, Y_2, \ldots, Y_n)$ be a basis of $\mathscr{S}$. For $t \in \mathbb{R}$, we denote by $M(t)$ the matrix whose columns are $Y_1(t), \ldots, Y_n(t)$.
Show that the matrix $(M(t))^{-1} M(t+T)$ is independent of $t \in \mathbb{R}$.

Let $T > 0$ be a real number and $n \in \mathbb{N}^*$ be a natural integer. Let $A : \mathbb{R} \rightarrow \mathscr{M}_n(\mathbb{C})$ be a continuous application on $\mathbb{R}$ and $T$-periodic. We consider the differential system $$X'(t) = A(t) X(t) \tag{2}$$ where $X$ is a function from $\mathbb{R}$ to $\mathbb{C}^n$, of class $\mathscr{C}^1$ on $\mathbb{R}$. Let $\mathscr{S}$ be the space of solutions in $\mathscr{C}^1(\mathbb{R}, \mathbb{C}^n)$ of (2). Let $(Y_1, Y_2, \ldots, Y_n)$ be a basis of $\mathscr{S}$. For $t \in \mathbb{R}$, we denote by $M(t)$ the matrix whose columns are $Y_1(t), \ldots, Y_n(t)$.
Deduce that there exists $B \in \mathscr{M}_n(\mathbb{C})$ such that: $$\forall t \in \mathbb{R}, \quad M(t+T) = M(t) \exp(TB)$$

Let $T > 0$ be a real number and $n \in \mathbb{N}^*$ be a natural integer. Let $A : \mathbb{R} \rightarrow \mathscr{M}_n(\mathbb{C})$ be a continuous application on $\mathbb{R}$ and $T$-periodic. We consider the differential system $$X'(t) = A(t) X(t) \tag{2}$$ where $X$ is a function from $\mathbb{R}$ to $\mathbb{C}^n$, of class $\mathscr{C}^1$ on $\mathbb{R}$. Let $\mathscr{S}$ be the space of solutions in $\mathscr{C}^1(\mathbb{R}, \mathbb{C}^n)$ of (2). Let $(Y_1, Y_2, \ldots, Y_n)$ be a basis of $\mathscr{S}$. For $t \in \mathbb{R}$, we denote by $M(t)$ the matrix whose columns are $Y_1(t), \ldots, Y_n(t)$. There exists $B \in \mathscr{M}_n(\mathbb{C})$ such that $M(t+T) = M(t)\exp(TB)$ for all $t \in \mathbb{R}$.
Deduce that there exists an application $Q : \mathbb{R} \rightarrow \mathrm{GL}_n(\mathbb{C})$ continuous on $\mathbb{R}$ and $T$-periodic such that $$\forall t \in \mathbb{R}, \quad M(t) = Q(t) \exp(tB)$$ (This identity is called the normal form of the matrix $M$).

Let $T > 0$ be a real number and $n \in \mathbb{N}^*$ be a natural integer. Let $A : \mathbb{R} \rightarrow \mathscr{M}_n(\mathbb{C})$ be an application continuous on $\mathbb{R}$ and $T$-periodic. We consider the differential system $$X'(t) = A(t) X(t) \tag{2}$$ where $X$ is a function from $\mathbb{R}$ to $\mathbb{C}^n$, of class $\mathscr{C}^1$ on $\mathbb{R}$. Let $\mathscr{S}$ be the space of solutions in $\mathscr{C}^1(\mathbb{R}, \mathbb{C}^n)$ of (2). Let $(Y_1, Y_2, \ldots, Y_n)$ be a basis of $\mathscr{S}$. For $t \in \mathbb{R}$, we denote by $M(t)$ the matrix whose columns are $Y_1(t), \ldots, Y_n(t)$.
Show that for every real number $t$, $M(t) \in \mathrm{GL}_n(\mathbb{C})$ and $M'(t) = A(t) M(t)$.