Solving Polynomial Question with Gauss Method

[iPanos]

Hey guys,

I am new to this forum and I have just found you after some long and unsuccessful research on the following question:

Homework Statement

The question is a combined matrix and polynomial question. First I am given the following matrix A:

2 1 1
5 4 -3
2 1 3

-1st sub-question is to calculate the characteristic polynomial of A, P(x).
-2nd sub-question is to find one eigenvalue for A, using Gauss method.
-3rd sub-question is to find a non-zero eigenvector for A, corresponding to the eigenvalue found in the second sub-question.

2. The attempt at a solution

I have solved the problem easily but for the second sub-question I did not use the requested method (Gauss). I found the roots of P(x) via the trinomial formulas and proceeded to sub-question 3.

The reason I did this is because I have no knowledge of some Gauss method for factoring polynomials, neither can I find it in google. This particular matrix has no integer roots, so it was a little complicated to just find one root by trying numbers that divide the constant.

So, my question is, how can Gauss get involved in this exercise?

To save you some time the characteristic polynomial is:
P(x)=-x^3+9x^2-22x+6
and its 3 real roots are:
4.3445...
0.3109...
4.3445...

Any help will be greatly appreciated...

 

[LCKurtz]

Not directly in response to your question, but Maple gives only one real root, your .3109.. one. Your 4.3445... ones are just the real part of the other complex conjugate pair.

 

[iPanos]

Not directly in response to your question, but Maple gives only one real root, your .3109.. one. Your 4.3445... ones are just the real part of the other complex conjugate pair.

Thanks for this, I noticed a little later...

Still working on this problem...