Find nondiagonalizable matrix A such that matrix $A^2-6A$ is diagonizible.
Simpliest nondiagonizable matrix is a Jordan block 2 by 2 $\begin{pmatrix} a & 1\\ 0 & a \end{pmatrix}$.
$\begin{pmatrix} a & 1\\ 0 & a \end{pmatrix} \cdot \begin{pmatrix} a & 1\\ 0 & a \end{pmatrix} -6\cdot\begin{pmatrix} a & 1\\ 0 & a \end{pmatrix}=\begin{pmatrix} a^2-6a & 2a-6\\ 0 & a^2-6a \end{pmatrix}$ is also nondiagonizable.
Is it possible to find such a matrix?