Question

6: Problem 5 Next Prev Up (1 pt) The matrix [ 55 35 A= 91 -57 has complex eigenvalues, A1,2 = a ± bi, where a = and b = The corresponding eigenvectors are v1,2 ct di, where c ) and d ( ( ).

Answer 1

L-91-57 1-91 -37-4 Ans: The given malmix A15317 the characterstic squation of Ah IA-AJ) = 0 ie 1 -) (55-1) (-57-1)-(-91135) = 0 =) (55X-57) – 5574 577 + 3185 = 0 - -3135+28+ x + 3185=0 - 12+2.1 +50 = 0 1 1 - 2 1 14-200 - 216 x = -2 + 149 - 1 = -) +78 Hence the eigen veeter rook of A= { 1 old, = -1-7° F - complex Eigenvalue diz= atbi. then a=-1; 6=7 For : A=-1+7 (68) ch vector corresponding i ů given by [55-6H71) 35 A dI] =0 =) -57-(-1471 L-91 22 7 0,5794216) 5657 556-71 357 -91 -56-78] [ > R₂ (56-71) +91 (R1) - [56-73 21:07 R2 =) (56-79)x, + 35x2 = 0 Нек, й free vao jcbe set X2 = k then Xi = " Xi - - 35 k --35 (56771 hen 56-71 (56-7)(56+71)

K 35x7 i La 5x56 1-35x56 + (-35x71) 35x9IJ 567 2 56² 772 -1-56-7K de-1+7) Corresponding the ch.vectr. (X, Y2) *7---77-7567 Simillons d=-1-71 ch vector corresponding t=-1-7i are given by 156+71 35 [A- El-71) 73 x 56+7 1-91-56+71) R2 - 910R) +156+7;)& [ 56+71 35 maya. Pa 11-13 2 - (56+71) X; + 35x2=0 x - free variable x = -35k 3 Set X2=k) free variable X2 56+79 35*9) Hen 5647 = 35491 3185 =3185 Xis -35(56-77) K 7315-35(56-78) (56+71)(56-77). x = (-56 + 7 ch. veeter of d=-1-71 917 X) a 1x12 -56 = 12 - 1 x2o ko Eigenvech set kal 2 = ctdi - fc = ( -5 1 1, 12 (= (-2, 0))