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

• A. \({e^{{\lambda _1}t}}\alpha\)
• B. \({e^{{\lambda _2}t}}\beta\)
• C. \({e^{{\lambda _2}t}}\alpha\)
• D. \({e^{{\lambda _1}t}}\alpha + {e^{{\lambda _2}t}}\beta\)