In Exercises 8.9 we saw that a nonzero n n matrix A is nilpotent if m is the smallest
Chapter 10, Problem 30(choose chapter or problem)
In Exercises 8.9 we saw that a nonzero n X n matrix A is nilpotent if m is the smallest positive integer such that \(\mathbf{A}^{m}=\mathbf{0}\). Verify that \(\mathbf{A}=\left(\begin{array}{ccc}-1 & 1 & 1 \\ -1 & 0 & 1 \\ -1 & 1 & 1\end{array}\right)\) is nilpotent. Discuss why it is relatively easy to compute \(e^{\mathbf{A t}}\) when A is nilpotent. Compute \(e^{\mathbf{A t}}\) for the given matrix and then use (2) to solve the system \(\mathbf{X}^{\prime}=\mathbf{A X}\).
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