I have this differential equation system
\begin{align} x'(t) - 3 x(t) + y(t) - z(t) &= 0 \\ y'(t) - 2 x(t) - z(t) &= 0 \\ z'(t) - x(t) + y(t) - 2 z(t) &= 0 \end{align}
I wrote the equation in the form $X'=AX$. And the matrix $A$ has two eigenvalues, $\lambda= 1$ and $\lambda= 2$ with (multiplicity 2). I think I found the right eigenvectors $(0,1,1)$ for eigenvalue 1 and $(1,1,0)$ and $(1,0,-1)$ for eigenvalue 2. I hope I got this right.
The answer is: x -> Function[{t}, E^(2 t) (1 + t) C[1] - E^(2 t) t C[2] + E^(2 t) t C[3]], y -> Function[{t}, E^t (-1 + E^t + E^t t) C[1] - E^t (-1 + E^t t) C[2] + E^(2 t) t C[3]], z -> Function[{t}, E^t (-1 + E^t) C[1] - E^t (-1 + E^t) C[2] + E^(2 t) C[3]]}}
How do I write $e^{At}$? Please help me, Thank you, Klara