This question is from Teschl ODE and dynamical systems. Problem 9.2
Let $A$ be a real-valued two by two matrix and let
$$χ_A (z) = z^2 − T z + D = 0,$$
$T = tr(A)$, $D = \det(A)$, be its characteristic polynomial. Show that A is hyperbolic if $T D \neq 0$. Moreover, $A$ is asymptotically stable if and only if $D > 0$ and $T < 0$. (Hint: $T = α_1 + α_2$ , $D = α_1 α_2 $.)
Let $A$ be a real-valued three by three matrix and let $$χ_A (z) = z^3 − T z^2 + M z − D = 0 $$
be its characteristic polynomial. Show that $A$ is hyperbolic if $(T M − D)D \neq 0$. Moreover, $A$ is asymptotically stable if and only if $D < 0$, $T < 0$ and $T M < D$. (Hint: $T = α_1 + α_2 + α_3 , M = α _1 α_2 + α_2 α_3 + α_2 α_3 , D = α_1 α_2 α_3 , and $T M − D = (α_1 + α_2 )(α_1 + α_3 )(α_2 + α_3 )$ .)
I know how to get the stability part but i tried investigating $ b^2 - 4ac $ for the hyperbolic part but i cant seem to get it. Can someone help? Thanks