I am presenting homework for the first time and would appreciate anyone looking over my work to see if I made any blunders before I present to the class
The question is "Find the steady states of the following system of equations and determine the Jacobian of the system for these steady states, stating whether the steady state is unstable, a saddle point, or asymptotically stable" $x^{2}-y^{2}=0$ and $x(1-y)=0$ is my system of equations. I began by finding the intersection points
(0,0) and (1,1)
$J(0,0)=\begin{pmatrix} 0& 0\\ 1 &0 \end{pmatrix}$
$\lambda_{1,2}$=0 therefore the point (0,0) is a saddle point
And then for $J(1,1)=\begin{pmatrix} 2& -2\\ 0 &-1 \end{pmatrix}$
$\lambda_{1}=2$ and $\lambda_{2}=-1$
therefore the point (1,1) is unstable