Consider the 2 parameter family of linear systems
$$\frac{DY(t)}{Dt} = \begin{pmatrix} a & 1 \\ b & 1 \end{pmatrix} Y(t) $$
In the ab plane, identify all regions where this system posseses a saddle, a sink, a spiral sink, and so on.
I was able to get the eigenvalues as $$\lambda = \frac{a+1}{2} \pm \frac{\sqrt{(a+1)^2 - 4(a-b)}}{2}$$
but need help in finding the sink and source.
I got the spiral sink as: if $a \lt -1$
spiral source if $a \gt -1$
and center if $a = -1$
Can someone check this?