$$ \left\{ \begin{array}{c} x' = 2x-3y-5z \\ y' = x+4y+z \\ z' = 2x+5z \end{array} \right. $$
Now, the roots of the characteristic polynomial are $\lambda_1 = \lambda_2 = 3$, and $\lambda_3=1$
This confuses me. In a previous exercise I had that the root was 1 with a multiplicity of three (three roots) so I had that the solution would simply look like this:
$$\begin{bmatrix}At+B\\Ct+D\\Et+F\end{bmatrix}e^t$$
Where I would have to express three constants in terms of the other three.
(The polynomials were of the first order, matrix rank was 1, I had three equations and the multiplicity was three)
But I can't do that here, right?
I thought that the solution would be like this:
$$\begin{bmatrix}At+B\\Ct+D\\Et+F\end{bmatrix}e^{3t} + C_1\begin{bmatrix}-6\\1\\3\end{bmatrix}e^t$$
I don't know if this is the correct way, because when I express the constants in terms of the other three, I already get $C_1$, $C_2$, $C_3$. Looking at my solution, I'd have four constants!
Can anyone help me in getting the correct form of the solution?