Huge polynomial eigenvalue problems

[K.I.L.E.R]

OMG!!! Expanding this is the shet.

Try a 4x4 matrix with some equation and get the eigenvectors for it.
Seriously this is some huge polynomial fun.

Polynomial of degree 4 is really cool to work out. Seriously.
I'm working out the determinant of a 3x3(degree 3 poly) matrix that is singular you cannot use Cramer's law on it so you have to work out the determinant the long way.

Seriously check out the expansion of the equations on this thing. A few pages of working out ought to do it.
Tell me there's a faster way of doing this? C'mon you maths geeks, tell me.

 

[bloodbob]

Use a computer

 

[K.I.L.E.R]

Defeats the purpose of doing the whole thing by hand for self learning.
I just need a shortcut.

Use a computer

 

[Fred]

Look for zeros, and expand it out intelligently.

Other than that, look for symmetry within the matrix, or something that looks familiar (say if its an antisymmetric matrix). Other than that, theres no real way out of it, just do it and try not to fall asleep. I think I was given a 7*7 or larger matrix once as a homework problem, needless to say I had to use a computer to check myself at each step, b/c the ease of making an error starts getting large.

There are other ways of doing it (for instance trying to arrange it so you can multiply the diagonal entries), but in general its not clear which way is faster

 

[K.I.L.E.R]

That's insane.
3x3 is bad enough but 7x7?