Consider the systemx = Ax =1112 1 1324 x. (i)(a) Show that r = 2 is an eigenvalue of

Chapter 7, Problem 18

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Consider the systemx = Ax =1112 1 1324 x. (i)(a) Show that r = 2 is an eigenvalue of algebraic multiplicity 3 of the coefficient matrixA and that there is only one corresponding eigenvector, namely,(1) =011 .(b) Using the information in part (a), write down one solution x(1)(t) of the system (i).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 theequations(A 2I) = 0, (A 2I) = .Since has already been found in part (a), solve the second equation for . Neglect themultiple 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 (i).(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 , againneglecting the multiple of (1) that appears. Then write down a third solution x(3)(t) of thesystem (i).(e) Write down a fundamental matrix (t) for the system (i).(f) Form a matrix T with the eigenvector (1) in the first column and the generalizedeigenvectors and in the second and third columns. Then find T1 and form the productJ = T1AT. The matrix J is the Jordan form of A.