Sketch solution of differential equation $$x'=\begin{pmatrix} \frac{4}{3} && \frac{2}{3}\\\frac{1}{3}&&\frac{5}{3}\end{pmatrix}x$$
From this I get the solution (by calculating the eigenvectors + values) to be $\begin{pmatrix}a\\a\end{pmatrix}e^{2t}+\begin{pmatrix}2b\\-b\end{pmatrix}e^t$.
Now actually the question was to just sketch the image of all solutions. What does this mean? Can I sketch the solutions without calculating the solution from just seeing the matrix?
The solutions are just lines? I mean I don't know what I have to sketch?
Is the idea to just shade the area between the two lines $(1,1)t$ and $(2,-1)t$?

We can plot a representative example of all solutions using the phase portrait.
We can then plot several solution curves for varying initial conditions for your system on top of the phase portrait.
Doing this results in (each color represents a different initial condition and there are 32 IC curves represented):

