Characteristic equation $|A-\lambda I|=0$
$\lambda_1=1+i \implies \lambda_2=1-i $
$\because$ all diagonal elements=0, $\Rightarrow \text{trace}=0$
Trace = sum of eigen values=0
$\implies\lambda_1+\lambda_2+\lambda_3=0$
$\implies (1+i)+(1-i)+\lambda_3=0$
$\implies \lambda_3=-2$
$|A|=\text{product of eigen values}=(1+i)(1-i)(-2)$
$\implies |A|=-4$