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?
