Diagonalizing matrix question for a ${M_{4x4}}$

Question

\begin{bmatrix} 1 & 1 & 1 & 1 \\ 0 & 0 & 2 & 0 \\ 0 & 2 & 0 & 0 \\ 3 & -1 & -1 & -1 \\ \end{bmatrix}

Determine if this 4x4 matrix is diagonalizable. How would you guys find the characteristic polynomial of this?

Answer 1

It's fairly painless to use the definition.

$$ p(\lambda) = \det \left( \mathbf{A} -\lambda \mathbf{I} \right) $$

You will end up at $$ p(\lambda) = \left( \lambda^{2} - 4 \right)^{2} $$

The solution is rather long - do you need that?