Gershgorin Circle theorem- implications

Question

(I am considering only real matrices)

Does only hold that if the area of all Gershgorin Circles is positiv $\Rightarrow$ the Matrix is positiv definit (trivial)

or does also follow the vice versa

the Matrix is positiv definit $\Rightarrow$ the area of all Gershgorin Circles is positiv

Answer 1

The reverse direction does not hold: $$ A=\pmatrix{ 1 & 2\\ 2 & 10} $$ is positive definite, but the Gershgorin circle for the first row contains numbers with negative real part.