Exercise 104

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 $A$.