After "This gives:", you need to check your math. I see (1-\lambda)^2(-1-\lambda)-4(1-\lambda)-4(1-\lambda)
To find eigenvectors, see if this resource helps: http://www.sosmath.com/matrix/eigen2/eigen2.html
Essentially, you will set up A \left(\begin{matrix} x \\ y \\ z...
The vector corresponding to NW is \hat{r}=\{-1/\sqrt{2},1/\sqrt{2}\}. Then the instantaneous rate of change in that direction, starting from point (8,6) is: \vec{\nabla}A(8,6)\cdot \hat{r} = ?
Great, so you can either state these in vector format: \hat{r}=-(3/5)\hat{x}+(4/5)\hat{y} (and similarly for -\hat{r}), or in angular format as measured from the positive x-axis.
See if this contour plot of A(x,y) helps. The blue vector represents the unit vector pointed in the direction of \vec{\nabla}A(x,y). The black vectors represent unit vectors in the direction of no ascent - they correspond to each \hat{r} you need to find.
I'm just throwing a guess in here, but if your goal is to find the velocity of something which is under the influence of a non-constant gravitational field, integrating its acceleration is not the easiest way to go. You would typically apply conservation of energy between initial and final...
The length of \vec{r} is of no concern, only its direction matters. In fact, here's what you really need to solve: \vec{\nabla}A(8,6)\cdot\hat{r}=0. Is this less ambiguous?