Preparing for entrance exams, I am in need of finding the spectra of the following matrices the most effectively. Anyone up for helping me find the best ways?
Matrix 1 | Matrix 2 | Matrix 3
The current technique I resort to is to use the multilinearity of the determinant columns and then Sarrus for the 3x3, cofactors for 4x4 and higher. Still, e.g. finding the spectrum of the first matrix takes quite some time.
Gauss' Elimination Method appears slow here, but maybe does not have to be? Decompositions by the linearity of e.g. the first matrix' columns lesser than four-fold also turn into mindless coefficient calculations.
I have heard a few times of techniques such as eigenvalue guessing. Would anyone kindly care to explain? (Possibly also other such techniques?) How exactly do you use the knowledge of the eigenvalue to simplify the calculations here?
By Cayley-Hamilton? Or through factoring out the polynomial root cleverly somehow?
Thanks.