Chapter 8.1, Problem 25E

To determine

To prove: The general solution of the homogeneous linear system $X^{\prime }=\left(\begin{array}{ccc}0& 6& 0\\ 1& 0& 1\\ 1& 1& 0\end{array}\right)X$ on the interval $\left(-\infty ,\infty \right)$ is $X=c_1\left(\begin{array}{c}6\\ -1\\ -5\end{array}\right)e^{-t}+c_2\left(\begin{array}{c}-3\\ 1\\ 1\end{array}\right)e^{-2t}+c_3\left(\begin{array}{c}2\\ 1\\ 1\end{array}\right)e^{3t}$.