Determine a fundamental matrix of a vertorial ODE. I must determine a fundamental matrix of the vectorial ODE
$\dot{x}(t) =  \begin{bmatrix}
1 & 4 \\
3 & 2 
\end{bmatrix} x(t)$.
In what form is a fundamental matrix? Should I just solve the ODE with eigenvalues and eigenvectors to compute the general solution? What is asked from me here?
 A: A fundamental matrix $\Phi(t)$ is a matrix $\Phi:I\rightarrow \mathbb{R}^{n\times n}$ such that $$\boxed{\Phi'(t)=\begin{pmatrix}1 & 4\\ 3 & 2\end{pmatrix}\Phi(t) \text{ and } \text{det}(\Phi(t))\neq 0, \forall t\in I}$$ For systems of differential equations $x'=Ax$ such that the matrix $A$ is constant (as in your case), then it can be proved that $$\Phi(t)=e^{tA}:=\sum_{n=0}^{\infty}\frac{t^nA^n}{n!}$$
The problem comes when the matrix $A$ is not diagonalized. In that case it must be first diagonalized and then compute the series. Suppose that $A=P^{-1}DP$, where $D$ is a diagonal matrix, then
$$\sum_{n=0}^{\infty}\frac{t^nA^n}{n!}=\sum_{n=0}^{\infty}\frac{t^n(P^{-1}DP)^n}{n!}=\sum_{n=0}^{\infty}\frac{t^nP^{-1}D^nP}{n!}=P^{-1}\sum_{n=0}^{\infty}\frac{t^nD^n}{n!}P$$ So, by using the definition of the exponential matrix this is the same as saying that $$e^{tA}=P^{-1}e^{tD}P$$ so it is essential for our problem to know how to compute $e^{tD}$ where $D$ is a diagonal matrix. But this is easy since $$\begin{pmatrix}1 & 4\\3 & 2\end{pmatrix}=\begin{pmatrix}-\frac{3}{7} & \frac{3}{7}\\ \frac{3}{7} & \frac{4}{7}\end{pmatrix}^{-1}\begin{pmatrix}-2 & 0\\0 & 5\end{pmatrix}\begin{pmatrix}-\frac{3}{7} & \frac{3}{7}\\ \frac{3}{7} & \frac{4}{7}\end{pmatrix}$$ and $$e^{tD}:=\sum_{n=0}^{\infty}\frac{t^nD^n}{n!}=I_2+tD+\frac{t^2D^2}{2!}+\dots=\begin{pmatrix}e^{-2t} & 0\\ 0 & e^{5t}\end{pmatrix}$$
So, $$\Phi(t)=\begin{pmatrix}-\frac{3}{7} & \frac{3}{7}\\ \frac{3}{7} & \frac{4}{7}\end{pmatrix}^{-1}\begin{pmatrix}e^{-2t} & 0\\ 0 & e^{5t}\end{pmatrix}\begin{pmatrix}-\frac{3}{7} & \frac{3}{7}\\ \frac{3}{7} & \frac{4}{7}\end{pmatrix}$$
to convince yourself that this is indeed a fundamental matrix for the system, try and see if it satisfies the definition of such a matrix.
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