[Math] Positive symmetric matrices and positive-definiteness

Question

Is a symmetric real matrix with diagonal entries strictly greater than $1$ and off-diagonal entries positive but strictly less than $1$ necessarily positive-semidefinite?

Answer 1

Nope. Just playing around with my computer, I found the matrix

[$\frac{11}{10}$ $\frac{1}{100}$ $\frac{99}{100}$]

[$\frac{1}{100}$ $\frac{11}{10}$ $\frac{99}{100}$]

[$\frac{99}{100}$ $\frac{99}{100}$ $\frac{11}{10}$]

with determinant $\frac{-25179}{31250}$.

Is this perhaps a misremembering of the definition of a diagonally dominant matrix?