(A[k_]={{1,-k},{k,1-(k^2)}})//MatrixForm
(x0={{1},{2}})//MatrixForm

Manipulate[
ListPlot[Table[MatrixPower[A[k],t,Flatten[x0]],{t,0,50,1}],AspectRatio->1,PlotRange->{{-35,35},{-35,35}}],
{k,-1.99,1.99,0.05}]