Consider the system x _ = Ax = 1 1 1 2 1 1 3 2 4 x. (38) a. Show that r = 2 is an

QUESTION:

Consider the system x _ = Ax = 1 1 1 2 1 1 3 2 4 x. (38) a. Show that r = 2 is an eigenvalue of algebraic multiplicity 3 of the coefficient matrix A and that there is only one corresponding eigenvector, namely, (1) = 01 1 . b. Using the information in part a, write down one solution x(1) (t) of the system (38). There is no other solution of the purely exponential form x = ert . c. To find a second solution, assume that x = te2t + e2t . Show that and satisfy the equations (A 2I) = 0, (A 2I) = . Since has already been found in part a, solve the second equation for . Neglect the multiple of (1) that appears in , since it leads only to a multiple of the first solution x(1) . Then write down a second solution x(2) (t) of the system (38). d. To find a third solution, assume that x = t2 2 e2t + te2t + e2t . Show that , , and satisfy the equations (A 2I) = 0, (A 2I) = , (A 2I) = . The first two equations are the same as in part c, so solve the third equation for , again neglecting the multiple of (1) that appears. Then write down a third solution x(3) (t) of the system (38). e. Write down a fundamental matrix (t) for the system (38). f. Form a matrix T with the eigenvector (1) in the first column and the generalized eigenvectors and in the second and third columns. Then find T1 and form the product J = T1AT. The matrix J is the Jordan form of A. 1