Repeated eigenvalues

Let $A = \left[ \begin{smallmatrix} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \end{smallmatrix}\right]$.

1. What are the eigenvalues?
2. What is/are the defect(s) of the eigenvalue(s)?
3. Find the general solution of $\vec{x}\,' = A\vec{x}$.

Let $A = \left[ \begin{smallmatrix} 1 & 3 & 3 \\ 1 & 1 & 0 \\ -1 & 1 & 2 \\ \end{smallmatrix}\right]$.

1. What are the eigenvalues?
2. What is/are the defect(s) of the eigenvalue(s)?
3. Find the general solution of $\vec{x}\,' = A\vec{x}$.

Let $A = \left[ \begin{smallmatrix} 2 & 0 & 0 \\ -1 & -1 & 9 \\ 0 & -1 & 5 \end{smallmatrix}\right]$.

1. What are the eigenvalues?
2. What is/are the defect(s) of the eigenvalue(s)?
3. Find the general solution of $\vec{x}\,' = A\vec{x}$.

Let $A = \left[ \begin{smallmatrix} a & a \\ b & c \end{smallmatrix}\right]$, where $a$, $b$, and $c$ are unknowns. Suppose that $5$ is a doubled eigenvalue of defect 1, and suppose that $\left[ \begin{smallmatrix} 1 \\ 0 \end{smallmatrix}\right]$ is a corresponding eigenvector. Find $A$ and show that there is only one such matrix $A$.