Let A be a 3x3 matrix.

char poly(A) = z^3 -Tz^2 + Mz - D = 0

Show that all eigenvalues of A have nonzero real part if (TM - D)D /= 0

Show that all eigenvalues of A are < 0 if D < 0, T < 0, and TM < D

T = tr(A) , D = det(A), and there's not a given definition for M.

If a1, a2, a3 are eigenvalues of A, I found that

T = a1 +a2 + a3

M = a1*a2 + a2*a3 + a1*a3

D = a1*a2*a3

TM - D = (a1 + a2)*(a1 + a3)*(a2 + a3)

I'm not sure where to go from here. Any help would be greatly appreciated!