GATE Electrical Engineering (EE) 2016 Shift 2 Solved Paper

Consider a linear time invariant system $\dot{\mathit{x}}=\mathit{A}\mathit{x}$ with initial condition $\mathit{x}\left(0\right)$ at $\mathit{t}=0$. Suppose $\mathit{\alpha }$ and $\mathit{\beta }$ are eigenvectors of $\left(2\times 2\right)$ matrix A corresponding to distinct eigenvalues ${\mathit{\lambda }}_1\mathit{a}\mathit{n}\mathit{d}{\mathit{\lambda }}_2$ respectively. Then the response $\mathit{x}\left(\mathit{t}\right)$ of the system due to initial condition $\mathit{x}\left(0\right)=\mathit{\alpha }$ is