• Show that the roots, x , of

    x^2 + a x + b =0

    where a,b are real have negative real parts if and only if both a,b are positive.

  • Show that the characteristic polynomial for a 2 x 2 real matrix, M has the form:

    x^2 + a x + b =0

    where -a is the trace of M and b is the determinant of M .

  • Use the above to exercises to prove that all solutions to

    X' = M X

    decay to 0 as t goes to infinity if and only if the determinant of M is positive and the trace is negative.