Problem 11.5. Find all possible values of the unknown constants given the condition.
1. $A = \begin{bmatrix} 0 & a \\ 1 & b \end{bmatrix}$ and the eigenvalues are 4, 7.
2. $A = \begin{bmatrix} 0 & 0 & a \\ 1 & 0 & b \\ 0 & 1 & c \end{bmatrix}$ and the eigenvalues are 2, 3, 4.
3. $A = \begin{bmatrix} 2 & -2 & 0 \\ -2 & x & -2 \\ -2 & -2 & 0 \end{bmatrix}$, $B = \begin{bmatrix} 2 \\ 2 \\ y \end{bmatrix}$, and A, B have the same characteristic polynomial.
4. $A = \begin{bmatrix} 2 & 1 & 1 \\ 1 & 2 & 1 \\ 1 & 1 & a \end{bmatrix}$ is invertible, and $x = \begin{bmatrix} 1 \\ b \\ 1 \end{bmatrix}$ is an eigenvector of A.
5. $A = \begin{bmatrix} 3 & 2 & -2 \\ -k & -1 & k \\ 4 & 2 & -3 \end{bmatrix}$ is diagonalizable.
