Consider a linear time invariant system $\dot{x}=Ax$ with initial condition $x(0)$ at $t=0$. Suppose $\alpha$ and $\beta$ are eigenvectors of $(2 \times 2)$ matrix $A$ corresponding to distinct eigenvalues $\lambda_{1}$ and $\lambda_{2}$ respectively. Then the response $x(t)$ of the system due to initial condition $x(0)=\alpha$ is
- $e^{\lambda_{1}t}\alpha$
- $e^{\lambda_{2}t}\beta$
- $e^{\lambda_{2}t}\alpha$
- $e^{\lambda_{1}t}\alpha+e^{\lambda_{2}t}\beta$