Question

5. (a) (10 pts) Find the eigenvalues of A= 4 -5 8 0 3 0 -1 3 -2 (b) (6 pts) Find a basis for any one of eigenspaces of A (you may use any eigenvalue you have found in (a)).

Answer 1

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Hello friend !!! Happy to answer you 0 4 -5 8 A = 3 3 - 2 j To find, eigen values of matrix A, we need to find roots of the characteristic polynomial coresponding to A: Characteristic polynomial is given by, det (A-21) =0 il I is 3x3 lidentity mat oix) 22 (trace A) a? Sum of principal minors of order 2 92-1A1 #sumof + Here, I AI denotes determinant of A - O → 22 (4+3-2)22 +(+6=8+8) 1 - (141) 1 Tal = 4(-6) 1(-24) - 24+24 ... foom eq? ☺ 2²-522+6 (125)+6) =O = 0

2 co 72-57 +6 = 6 (2-2) -3) - 2= 0,2,3 Thus el correspond matrix 0,2,3 to gen value A are

Only one vector Now, to find basis corresponding any one eigen space of A. | NOTE since, each eigen value is repeated only once. Therefore, corresponding eigen space to each eigen valúe contains → It's basis consists only me element. (comesponding each eigen) To find eigen vector, Consider, [A -25] (x) = 0 for A=0 0 -1 value 1 3 3 -5 le 8 o -2 O 24 4 -5 3 I ooo y 8 -2 NO

R₂ R3 - Q Rp 4 一5 3 1111 R2 4 ) R₂-RI -5 Vio O 42 z = 0 124 +7Z 20 3 ने 2= 4 o af -73 12 X # 7 - 1 2 3 3 z \. is constant. Let z za 14 → -a -7/12 z corresponding Thus basis of eigen space to eigen value 720 is 14 -7/12