Chapter 8.2, Problem 54E

Show that the 5 × 5 matrix

$A=\left(\begin{array}{ccccc}2& 1& 0& 0& 0\\ 0& 2& 0& 0& 0\\ 0& 0& 2& 0& 0\\ 0& 0& 0& 2& 1\\ 0& 0& 0& 0& 2\end{array}\right)$

has an eigenvalue λ₁ of multiplicity 5. Show that three linearly independent eigenvectors corresponding to λ₁ can be found. Consider the 5 × 5 matrix given in Problem 33. Solve the system X′ = AX without the aid of matrix methods, but write the general solution using matrix notation. Use the general solution as a basis for a discussion of how the system can be solved using the matrix methods of this section. Carry out your ideas.