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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?

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  • $\begingroup$ What's this Wolfram Alpha? $\endgroup$ – user3767495 Nov 27 '18 at 5:35
  • $\begingroup$ @Moo-Thanks for suggessting the tool. Please answer my second query. $\endgroup$ – user3767495 Nov 27 '18 at 5:50
  • $\begingroup$ @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. $\endgroup$ – user3767495 Nov 27 '18 at 5:56
  • $\begingroup$ But I have a good intuition that Determinant don't change with row transformations. $\endgroup$ – user3767495 Nov 27 '18 at 5:59

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