Chapter 8.4, Problem 26E

A matrix A is said to be nilpotent if there exists some positive integer m such that A^m = 0. Verify that

$A=\left(\begin{array}{l}\begin{array}{ccc}-1& 1& 1\end{array}\\ \begin{array}{ccc}-1& 0& 1\end{array}\\ \begin{array}{ccc}-1& 1& 1\end{array}\end{array}\right)$

is nilpotent. Discuss why it is relatively easy to compute e^At when A is nilpotent. Compute e^At and then use (1) to solve the system X′ = AX.