Chapter 9.5, Problem 39E

a. Show that the matrix

$\text{A}=\left[\begin{array}{ccc}2& 1& 1\\ 1& 2& 1\\ -2& -2& -1\end{array}\right]$

has the repeated eigenvalue $r=1$ with multiplicity 3 and that all the eigenvectors of A are of the form $\text{u}=s\text{col}\left(-1,\text{ 1, 0}\right)+v\left(-1,\text{ 0, 1}\right)$.

b. Use the result of part (a) to obtain two linearly independent solutions of the system $X^{\prime }=\text{A}X$ of the form $X_1\left(t\right)=e^t{u}_1$ and $X_2\left(t\right)=e^t{u}_2$.

c. To obtain a third linearly independent solution to $X^{\prime }=\text{A}X$

try $X_3\left(t\right)=t{e}^t{u}_3+e^t{u}_4$. [Hint: show that $u_3$ and $u_4$ must satisfy $\left(\text{A}-I\right)u_3=0,\left(\text{A}-I\right)u_4=u_3$.

Choose $u_3$ an eigenvector of A, so that you can solve for $u_4$.]

d. What is ${\left(A-I\right)}^2{u}_4$?