Suppose $A \in M_n(\mathbb R)$ is a stable matrix, i.e., all eigenvalues are on the left open half plane of $\mathbb C$. If in particular, all the Gershgorin disks $\Gamma_j$ corresponding to rows $j=2, \dots, n$ are on the left open half plane and the disk $\Gamma_1$ is not contained in the left open half plane. Will the matrix remain stable if we decrease the entry $A_{11}$? For example, if $A = \pmatrix{0.95 & -1 \\ 2 & -2.05}$, the matrix has eigenvalues $-0.05, -1.05$ and is stable. The Gershgorin disks are $\Gamma_1 = \{z: |z-0.95| \le 1\}$ and $\Gamma_2 = \{z: |z+2.05| \le 2\}$. When I run simulations, it seems that $A$ remains stable.
A related question is this question I asked before and above example is from the answer by @loup blanc there.