Is there a fast way of getting the eigenvalues of this 3x3 matrix?

Question

Matrix: \begin{bmatrix} 4& -4 &4 \\ -3& 3 &5 \\ 1& -1&9 \end{bmatrix}

I want to check whether this matrix is diagonalizable and thus I want to calculate the eigenvalues. I know that this is possible with cofactor expansion and I have tried it and came to the right answer, but is there any FASTER way of getting eigenvalues of such matrix?

Answer 1

Since the first two columns are multiples of each other, so $\lambda=0$ is definitely an eigenvalue. Also $\text{trace}=16$, so sum of the other two eigenvalues is $16$.

Also observe that (by adding the second and third column) $$A\begin{bmatrix}0\\1\\1\end{bmatrix}=8\begin{bmatrix}0 \\ 1\\ 1\end{bmatrix}.$$

Thus $\lambda=8$ is a repeated eigenvalue (using the trace idea).