For an $n$ by $n$ symmetric matrix $M$, why does $Tr(M)^2/Tr(M^2)>n-1$ imply that M is positive or negative definite?

Question

Let $M$ be an symmetric square matrix of size $n$. I am trying to prove that $Tr(M)^2/Tr(M^2)>n-1$ is a sufficient condition for proving that $M$ is positive or negative definite. If in addition $Tr(M)>0$, then $M$ will be positive definite, and if $Tr(M)<0$ then $M$ will be negative definite. See also: https://en.wikipedia.org/wiki/Definite_matrix#Trace.