Find the Fundamental Matrix associated with $A$ 
Let $\frac{dy}{dt} = Ax + f(t),$ where $f(t)= \langle 2e^{-t},4e^{-t}\rangle$
and $x_1(t)= \langle e^{3t},2e^{3t}\rangle$, $x_2(t)= \langle -e^{-t},2e^{-t}\rangle$ are the fundamental solutions of
$\frac{dx}{dt}=Ax$.

I am really unsure how to find the fundamental matrix associated with $A$ with the given information. Thanks!
 A: Let's recall the definition of such a matrix.

Given a linear homogeneous system of differential equations of the form $$x'(t)=A(t)x(t)$$
with $A:I\rightarrow\mathcal{L}(\mathbb{R}^n,\mathbb{R}^n)$ such that $t\mapsto A(t)$ and $x(t)\in\mathbb{R}^n$. $$$$
We say that a matrix $\Phi(t)$ is a fundamental matrix for the system defined above if and only if $$\Phi'(t)=A(t)\Phi(t) \hspace{16mm} (1)$$ and $$\text{det}(\Phi(t))\neq 0, \hspace{3mm}\forall t\in I \hspace{6mm}(2)$$

Let's illustrate what I mean with some examples.

Example 1:
Consider the following system $x'=Ax$ where $A$ is a constant-valued matrix of the form $$A=\begin{pmatrix}0 & 1\\-1 & 0\end{pmatrix}$$ Given, $$\Phi(t)=\begin{pmatrix}\text{cos}(t) & \text{sin}(t)\\ -\text{sin}(t) & \text{cos}(t)\end{pmatrix}$$ it is easy to verify that $\Phi(t)$ satisfies $(1)$ and $(2)$.
$$\Phi'(t)=\begin{pmatrix}-\text{sin}(t) & \text{cos}(t)\\ -\text{cos}(t) & -\text{sin}(t)\end{pmatrix}=\begin{pmatrix}0 & 1\\-1 & 0\end{pmatrix}\begin{pmatrix}\text{cos}(t) & \text{sin}(t)\\ -\text{sin}(t) & \text{cos}(t)\end{pmatrix}=A\Phi(t)$$ and $$\text{det}(\Phi(t))=1\neq 0$$ So we can say that $\Phi(t)$ is a fundamental matrix for the system defined above.

Example 2:
Consider the system $x'=A(t)x$, where $$A(t)=\begin{pmatrix}0 & 1\\ -\frac{2}{t^2} & \frac{2}{t}\end{pmatrix}, \hspace{3mm}I=\mathbb{R}\setminus \{0\}$$ Notice that this time the matrix $A(t)$ is not contant.
Given, $$\Phi(t)=\begin{pmatrix}t^2 & t\\ 2t & 1\end{pmatrix}$$ prove that $\Phi(t)$ is a fundamental matrix for the system $x'=A(t)x$.

So, now that we have illustrated a couple of examples we can start to solve your problem. Notice that on every example given, the columns of the matrix $\Phi(t)$ are linearly independent solutions of the system.
What I mean by that is, considering the second example, $\varphi_1(t)=(t^2,2t)$ and $\varphi_2(t)=(t,1)$ are linearly independent vectors $\forall t\in I$ such that $$\varphi_i'(t)=A(t)\varphi_i(t),\hspace{5mm}i=1,2.$$
So in a way, and this can be proved, we can show that a fundamental matrix is a matrix whose columns are linearly independent solutions of the homogeneous system of equations $x'=A(t)x$.
So, considering the homogeneous part of your problem $x'=Ax$, a fundamental matrix for it would be $$\boxed{\Phi(t)=\begin{pmatrix}x_1(t)^T & x_2(t)^T\end{pmatrix}=\begin{pmatrix}e^{3t} & -e^{-t}\\ 2e^{3t} & 2e^{-t}\end{pmatrix}}$$ notice that this matrix satisties the definition of a fundamental matrix.
Also, notice that we have not talked about the non-homogeneous case where the system is of the form $x'=A(t)x+f(t)$, that is because this functions $f(t)$ takes no part in finding a fundamental matrix for the system, since we must always consider the homogeneous case first $x'=A(t)x$.
Finally, if you are wondering the reason behind finding this matrix, I must say that this is the start point in finding the general solutions for any system of linear differential equations (homogeneous or not).
