Eigen Values of matrix

I usually have a problem working on below type of matrices, where I am damn sure I'm likely to make a calculation mistake.

$$\begin{bmatrix} 0&2&-2 \\ -12&-22&12 \\ -12&-22&10 \end{bmatrix}$$

It is asked to find the ratio of absolute maximum eigenvalue to the absolute minimum eigenvalue.

I start by finding the characteristic polynomial and finding the roots of that polynomial to help me reach the final answer, but midway I always make some calculation mistake.

Is there any better way to find eigenvalues of such matrices where chances of making error are less?

Also, I want to ask one more point. My characteristic polynomial is

$$\begin{vmatrix} -\lambda&&2&&-2\\-12&&-22-\lambda&&12\\-12&&-22&&10-\lambda \end{vmatrix}=0$$

Now to the above determinant can I apply row transformations of form $$R_i=R_i+kR_j$$, Am I guaranteed to get same characteristic polynomial, without any change?

Like I can see, I can do, $$R_2=R_2-R_3$$ Then I get

$$\begin{vmatrix} -\lambda&&2&&-2\\0&&-\lambda&&\lambda+2\\-12&&-22&&10-\lambda \end{vmatrix}=0$$

Still after doing such transformation, will get same Characteristic polynomial?

• What's this Wolfram Alpha? – user3767495 Nov 27 '18 at 5:35
• @Moo-Thanks for suggessting the tool. Please answer my second query. – user3767495 Nov 27 '18 at 5:50
• @Moo-that means I have to solve a full 3x3 determinant? I am asking so much because the exam I am preparing for requires questions to be solved under 3 minutes. – user3767495 Nov 27 '18 at 5:56
• But I have a good intuition that Determinant don't change with row transformations. – user3767495 Nov 27 '18 at 5:59