Chapter 9.5, Problem 37E

To determine

(a)

To prove:

The given matrix has the repeated eigenvalue $r=2$ and all the eigenvectors are of the form $\text{u}=s\text{col}\left(1,\text{ 0, 0}\right)$.

To determine

(b)

To find:

The nontrivial solution $X_1\left(t\right)$ of the system $X^{\prime }=\text{A}X$ for the given condition.

To determine

(c)

To find:

The second linear independent solution of the system $X^{\prime }=\text{A}X$ for the given condition.

To determine

(d)

To find:

The third linear independent solution of the system $X^{\prime }=\text{A}X$ for the given condition.

To determine

(e)

To prove:

The expression ${\left(A-2I\right)}^2{u}_2={\left(A-2I\right)}^3{u}_3=0$.