Show that A = $\left[\begin{array}{ccc} 0 &0 &0\cr -4 &2 &2\cr 6 &-3 &-3 \end{array}\right]$ and $B=\left[\begin{array}{ccc} -9 &-3 &-9\cr 15 &5 &15\cr 3 &1 &3 \end{array}\right]$ are similar matrices by finding an invertible matrix satisfying $A=P^{-1} B P$ (Hint: the matrices and are diagonalizable with the same eigenvalues.)
$P^{-1}=\left[\begin{array}{ccc} & & \cr & & \cr & & \end{array}\right]$ and $P=\left[\begin{array}{ccc} & & \cr & & \cr & & \end{array}\right]$
I started to find the eigenvalue and eigenvector of B, but I cannot get it right. What should I do next?
You can easily see that the null space for the matrices has dimension $2$ therefore $\lambda=0$ are $2$ eigenvalues and by the trace we can see that also $\lambda=-1$ is an eigenvalue.
Form here we can proceed to find eigenvectors and then find the matrix $P$.
For a check refer to