I'm having trouble solving the following question: Let A be a matrix with the following characteristic polynomial: $\,p(t)=t^3+2t^2-3t\,$.
Show that the matrix $\,A^2+A-2I\,$ is similar to the matrix$\begin{pmatrix}0&0&0\\0&4&0\\0&0&\!\!\!-2\end{pmatrix}$
I've already figured out that the eigenvalues of $A$ are $0,1,-3$, and that $\,A^2+A-2I\,$ equals to $\,(A+2I)(A-I)\,$. Also, the eigenvalues of $\,A+tI\,$ are the eigenvalues of $\,A+t\,$ ($\,+t\,$ to each eigenvalue). The only thing left now is the multiplication of the two matrices, what can I know about it? Is there another way to solve the question? Thanks!