It is given that 2 years' amount $\(\left(\mathrm{A}_2\right)\)$ is $\(\$ 900\)$ and 3 years' amount $\(\left(\mathrm{A}_3\right)\)$ is $\(\$ 1050\)$. This implies that 1 year's simple interest $\((\mathrm{SI})=\mathrm{A}_3-\mathrm{A}_2=\$ 150\)$.

$\(\therefore\)$ Principal (Initial amount) $\(=A_2-(2$ years' SI $)=900-300=\$ 600\)$.

Thus, the correct answer is $\(\$ 600\)$.