Eigenvalues; help with calculating characteristic polynomial roots

I need to find the eigenvalues of the linear transformation that the following matrix represent:

so I calculated the characteristic polynomial of A:

but I have no idea how to find its roots...

any ideas would be greatly appreciated.

TIA!

**edit

Is there another way to find the eigenvalues of A?

Re: Eigenvalues; help with calculating characteristic polynomial roots

There is some methods that work on a 3rd degree polynomial. You could use a factoration like

where are polynomial roots.

There is another method that is called Ruffini's method (or Briot-Ruffini's method)

Ruffini's rule - Wikipedia, the free encyclopedia, take a look!

Re: Eigenvalues; help with calculating characteristic polynomial roots

Hi bosabarbosa, thanks for the help.

I'm familiar with polynomial division, but it is not useful here, since i don't know what are the divisors to be begin with (to use the example from the wiki-link - I don't know what 'r' is).

Division by when 'r' isn't a root will produce remainder.

I'm looking for a way to find the roots without dealing with the polynomial's standard form ( ).

(maybe calculating the determinant in some other way will produce an easier-to-handle polynomial?)

Re: Eigenvalues; help with calculating characteristic polynomial roots

Quote:

Originally Posted by

**Stormey** I need to find the eigenvalues of the linear transformation that the following matrix represent:

so I calculated the characteristic polynomial of A:

You made a mistake calculating the determinant.

Re: Eigenvalues; help with calculating characteristic polynomial roots

I tried is some manipulations that seems to work fine:

doing again, we get:

Now, if you try to complete the square:

Factoring :

Now, you have a root, 4! Now, the rest is up to you. :D

Re: Eigenvalues; help with calculating characteristic polynomial roots

Thats brilliant man.

Thank you so much!