Diagonalizability of matrix A

Question

Show that A is not diagonalizable in the form of $PDP^{-1}$ given eigenvalues.

In this question, A is $\begin{pmatrix} 54 & -220 & 26 & -66 \\ 12 & -50 & 6 & -15 \\ 86 & -352 & 45 & -108 \\ 40 & -160 & 20 & -50\end{pmatrix}$ and

D (or the eigenvalues) is $\begin{pmatrix} -2 & 0 & 0 & 0 \\ 0 & 2 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & 1\end{pmatrix}$.

I just need some hints to get going, is it really inevitable to have to calculate the characteristic polynomial?

Answer 1

Hint: One basis-invariant quantity that is easy to compute is the trace.

Answer 2

If $A$ has really the eigenvalues $-2,2,-1$ and $1$, then $A$ is diagonalizable !

Thus check, what are the eigenvalues of $A$.