Computing eigenvalues with characteristic polynomial I have two questions about computing eigenvalues with the characteristic polynomial.  


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*Eigenvalues exist if and only if I can factor the polynomial?? For example, I know i can calculate the roots of $ t^2 - 3t + 3 $ but I would use a quadratic formula for that.  

*An exercise asks me to find the eigenvalues of a matrix  $$A = \left( \begin{array}{cc} 1 & 2 \\ 3 & 2 \end{array}\right)$$
Computing $\det(A-tI) $ I get $ t^2 - 3t + 3 $.
Computing $\det(tI -A) $ I get $ t^2 - 3t - 4 = (t-4)(t + 1) $
I didn't come up with the last solution, but I'm not even sure why it is correct to compute $\det(tI -A) $ instead of $\det(A-tI) $, and in which cases it is more convenient to do that.


Eigenvalues may be imaginary, if you calculate the characteristic polynomial and set it equal to zero to find the roots then just as with a quadratic which may have no real solutions the characteristic polynomial may have imaginary roots. Take a rotation matrix as an example Eigenvalues of a rotation
