System of first order linear differential equations 
The solutions of a homogeneous system of linear differential equations of the first order $$\frac{d \vec x}{d t} = A \vec x$$ are $$\vec x = e^{ \lambda t}  \vec v $$  where $ \lambda $ are the eigenvalues of A and $\vec v$ are the associated eigenvectors.
So the first and second options are true. As for the others I'm not sure.

Consider the matrix
\begin{align}
A =&\ 
\begin{pmatrix}
1 & -1 & 1 & 0\\
-1 & 2 & 0 & 1\\
1 & 2 & 0 & 0 \\
-2 & -1 & 0 & 0
\end{pmatrix}
\begin{pmatrix}
-2 & 0 & 0 & 0\\
0 & -1 & 0 & 0\\
0 & 0 & 0 & 0\\
0 & 0 & 0 & 0
\end{pmatrix}
\begin{pmatrix}
1 & -1 & 1 & 0\\
-1 & 2 & 0 & 1\\
1 & 2 & 0 & 0 \\
-2 & -1 & 0 & 0
\end{pmatrix}^{-1}\\
=&\ \frac{1}{3}\begin{pmatrix}
0 & 0 & 4 & 5\\
0 & 0 & -6 & -6\\
 0 & 0 & -2 & 2 \\
0 & 0 &-2 & -7
\end{pmatrix}
\end{align}
then it's clear that $v=(-1, 2 ,2, -1)^T$ is an eigenvector with eigenvalue $-1$. Likewise, $w=(1, -1, 1, -2)^T$ is an eigenvector with eigenvalue $-2$. However, 3 is not an eigenvalue of $A$ so $(A-3I)\mathbf{x} = \mathbf{0}$ has only a trivial solution.  It's also easy to check that the last statement is false. 
