Question

3. For each of the following statements decide if it is true or false. If it is true, prove it. If it is false, give an example for which it does not hold. (a) If is an eigenvalue of the (n, n)-matrix A, then 2 - 31+ 512 is an eigenvalue of 21_n - 3A + 5A2 (b) The complex vector V1 = (1 + 1,0,1) is an eigenvector of the matrix [ 2 0 -4 ] A= | 0 1 0 [ 2 0 -2 ] with eigenvalue 11 = 2i . (C) The eigenspaces of the matrix [ 20 -41 A = | 0 1 0 2 0 -2 are E(2i) = span {vı} E(-21) = span {v2} Ei = span {v3} with with with vi = (1 + i, 0, 1) V2 = (1 – 1,0,1) V3 = (0,1,0) Hint: use Part b). (d) The matrices [ 2 1 4] [100 A= 0 2 3 , and B = -1 4 0 , [004] [234] are not similar; i.e. there does not exist an invertible matrix C such that A = CBC-1, (e) If A is a diagonalizable matrix of size (4,4) with eigenvalues 1, 2, 3 and 4, then det(A) = 4!.

Answer 1

③ @ Let at a be an ei eigen ralue of A Then, there enist un eigen rector v. such that Ausav ij How @ I on valv. =) 218nne 2.0 - 0. Avaar. 7-3 Avs-314 22). Mr Arzand si Arv = AAN = AAU [from ①] = Arozore =) SAMU=572e . Mow, (22. dn = 3A + SAJ U = 2320"- 3 Ar + SAMO = 2.0 - 313+5 1 from ol (218 m - 3 A + SA") =(2-3x+52" he Land Hence 2-3x +57" is an eigen valine of 28 dn-3A +54" with eigen reefore

6 let (Ah, as, kg) " be an eigen reefor corr, to the rigen vakre zi Then 12-21 o-4 a/m . 10 1 2 0 -2-211 Then (2-21) % * - 40g =0, (+-22)% =0, Demo ) 214-(2+2)83 20, Now. (1-2)22 20 => N₂ 20. Take My =1, then on- (2+2i) 2002 - Hi. And Citi, o, 1) also satisfy (2-21) 2-493 20, hun Clte, 0, 1) is an eigen reetor corr, to ze. © Already proved. laid- eigen reefor corr. to the eigenvalue zi is (1+1,0,1). Thas Elzi) = span { vil with ",= (te, 0, 1). Now let (Y, M2 Yg)t be eigen ree for corr. to age. Then (atrio-4 lei o ltzio 11 | 2 or2t2i/l } Then (2+28)%, - 4% 20. (1+22)% =0. 29, -(2-2) 4₂20 . . " mi (1428) % 20 => Z2 20,

Take Yg=1. Then Y = lri . How to (t-1, o, ip ako satisfy (2+2ily = (2+21) y, +-4 Yg co, an das (lni, o,1) is the eigen vector corr, to the eigen value -zi Hence El-2) = span {N2 ya with Vz={1-1,0,14 How let (W., We, wat be eigen reefor corroto i; Then. ( 20-31. Ws) lo. Wi-4 W220, 20,- 3 W3 to this two These two equine imply that wia Wg ro. Take W₂ =1. Then (0,1,0) is an eigen rector corr-to1. Then Ed = span {NY or with V₂ = (0,1,0) @ Let A and B are similar. Tagsre. I am invertible matrix much that A = CBC a. Then det(At dettes dettB). But dettag=to land det ba Then A and B have same eigen value but age eigen valores of A are 60,2,2,4 and eigen valmes of B are 1,4,4.

© If Ain a diagonalizable matrix of size (4,4) with eigen values 1, 2, 3, 4, Toe Then there enist an invertsble matrix p and a diagonal matrix D whose diagonal elements are. 1, 2, 3,4 such that A = PD Pr! No det (A)= det (PBP") - det (1) = 1.2.3.4 1 Then, Edy!, Hence @ true. ② true @ true. @ false o true.