= Solution
{c}
The <Gershgorin circle theorem> states that every <eigenvalue> of $A$ lies in at least one disc
$$
D_i=\left\{z\in\mathbb C:|z-a_{ii}|\leq\sum_{j\ne i}|a_{ij}|\right\}.
$$
For a nonzero complex <eigenvector> $v$, choose an index $i$ with $|v_i|=\max_j|v_j|>0$. The $i$th row of $Av=\lambda v$ gives
$$
|\lambda-a_{ii}|\,|v_i|\leq\sum_{j\ne i}|a_{ij}|\,|v_j|
\leq\left(\sum_{j\ne i}|a_{ij}|\right)|v_i|.
$$
Dividing by $|v_i|$ proves membership in $D_i$. This is valid for real matrices even when the eigenvalues are complex.
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