linear 4×4 system of nonhomogeneous differential equations with repeated eigenvalues.

I'm trying to find the general solution of this system of linear Nonhomogeneous differential equation: $$X'=\begin{bmatrix} -2 &1 &1 &-1\\ -1 & -1 & 1 &0\\ 0 &0 &-1 &0\\ 0 &-1 &0 &0 \end{bmatrix}X + \begin{bmatrix} e^t \\ 2t \\ e^{-2t} \\ 0 \end{bmatrix} $$ as an aproach, I found the eigenvalue $-1$ with multiplicity of 4. And I found two eigenvectors: $$V^{(1)}=\begin{bmatrix} 1 \\ 0 \\ 1 \\ 0 \end{bmatrix}$$ and $$V^{(2)}=\begin{bmatrix} 0 \\ 1 \\ 0 \\ 1 \end{bmatrix}$$ so i got two answers for the Corresponding homogeneous equation: $$X^{(1)}=\begin{bmatrix} 1 \\ 0 \\ 1 \\ 0 \end{bmatrix}e^{-t}$$ and $$X^{(2)}=\begin{bmatrix} 0 \\ 1 \\ 0 \\ 1 \end{bmatrix}e^{-t}$$

since we have 2 eigenvectors and one eigenvalue with multiplicity of 4, I tried to find the other two answers by assuming them as $X=vte^{-t} + ηe^{-t}$ and I got:

$X^{(3)}=\begin{bmatrix} 0 \\ 1 \\ 0 \\ 1 \end{bmatrix}te^{-t} + \begin{bmatrix} 0 \\ 0 \\ 1 \\ 1 \end{bmatrix}e^{-t}$

$X^{(4)}=\begin{bmatrix} 0 \\ 1 \\ 0 \\ 1 \end{bmatrix}te^{-t} + \begin{bmatrix} -1 \\ -1 \\ 0 \\ 0 \end{bmatrix}e^{-t}$

I wanna know if the Answers I found are correct. if Yes, we have: $$X=\begin{bmatrix} e^{-t} &0 &0 &-e^{-t}\\ 0 & e^{-t} & te^{-t} &te^{-t}-e^{-t}\\ e^{-t} &0 &e^{-t} &0\\ 0 &e^{-t} &te^{-t}+e^{-t} &te^{-t} \end{bmatrix}$$ using the Variation of Parameters method, we have to inverse X to find the answer but A is singular $(detA=0)$ :(

where am I going wrong? can someone help me?

You assumed that the Jordan blocks for $-1$ were of sizes $\{2,2\}$ – they're in fact of sizes $\{3,1\}$: $$A=\begin{bmatrix}0&-1&1&1\\1&-1&0&0\\0&0&0&1\\1&0&0&0\end{bmatrix}\begin{bmatrix}-1&1&0&0\\0&-1&1&0\\0&0&-1&0\\0&0&0&-1\end{bmatrix}\begin{bmatrix}0&-1&1&1\\1&-1&0&0\\0&0&0&1\\1&0&0&0\end{bmatrix}^{-1}$$ So one homogeneous solution is a vector times $t^2e^{-t}$: $$X=\begin{bmatrix}1-t&t&t&-t\\-t+t^2/2&1-t^2/2&t-t^2/2&t^2/2\\0&0&1&0\\t^2/2&-t-t^2/2&-t^2/2&1+t+t^2/2\end{bmatrix}e^{-t}$$