Chapter 9.4, Problem 29E

In Problem 29-30, verify that $X\left(t\right)$ is a fundamental matrix for the given system and compute $X^{-1}\left(t\right)$. Use the result of Problem 28 to find the solution to the initial value problem.

$x^{\prime }=\left[\begin{array}{ccc}0& 6& 0\\ 1& 0& 1\\ 1& 1& 0\end{array}\right]x,x\left(0\right)=\left[\begin{array}{c}-1\\ 0\\ 1\end{array}\right]$

$X\left(t\right)=\left[\begin{array}{ccc}6e^{-t}& -3e^{-2t}& 2e^{3t}\\ -e^{-t}& e^{-2t}& e^{3t}\\ -5e^{-t}& e^{-2t}& e^{3t}\end{array}\right]$