Consider the 5 × 5 matrix given in Problem 31. Solve the system ' = 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.

Problem 31.

Show that the 5 × 5 matrix

has an eigenvalue λ₁ of multiplicity 5. Show that three linearly independent eigenvectors corresponding to λ₁ can be found.

2 1 00 0' 0 2 0 0 0 A = 0 0 2 0 0 0 0 0 0 2 1 0 0 0 0 2