
Well, if you know how to find the characteristic polynomial, then that's all
The roots of the characteristic polynomial are the eigenvalues. So find $\displaystyle \alpha$ such that there are indeed positive.
Characteristic polynomial  Wikipedia, the free encyclopedia
Positivedefinite matrix  Wikipedia, the free encyclopedia (but I'm afraid this article may not correspond to your level...I can't tell)
That's the part you need.Let M be an n × n [...] matrix. The following properties are equivalent to M being positive definite:
1. All eigenvalues λi of M are positive. [...]
Hmmm
A diagonal 2x2 matrix is in the form $\displaystyle \begin{pmatrix} \gamma & 0 \\ 0 & \delta \end{pmatrix}$
And the equivalent diagonal matrix to a given matrix will be formed by its eigenvalues. That is to say, in our case here, $\displaystyle \gamma$ and $\displaystyle \delta$ are the eigenvalues of A.
Did you find them ?
For the eigenvectors, it's better you have a look on the internet...
Try this : Pauls Online Notes : Differential Equations  Review : Eigenvalues & Eigenvectors (it's also explain how to get the eigenvalues)
Take the time to read it please