# Positive trace (all diagonal entries are positive) implies semipositive definite?

I am working on a matrix with all the diagonal entries are strictly positive while every other entry is strictly negative. This matrix is symmetric as well. I want to show that this matrix is semipositive definite. Since the trace is strictly positive, I know that the sum of the all eigenvalues (the roots of characteristic polynomial) is also strictly positive, but I am not sure whether each eigenvalues are nonnegative. What is the best way to show this matrix is semipositive definite?

It's not true. $$\begin{bmatrix}1 & -2\\-2 & 1\end{bmatrix}$$ is a simple counter example